Why do I have to put the substituted number in parentheses?

Because algebra hides multiplication signs and exponent bases. In 6x the multiplication is invisible, so substituting -2 without parentheses gives 6-2, which reads as subtraction. And in x^2, writing -2^2 makes the exponent apply only to the 2. Parentheses keep the meaning of the original expression intact, so they cost you nothing and prevent the two most common errors.

Is (-6)^2 the same as -6^2?

No. (-6)^2 means (-6)(-6) = 36, because the base is the whole quantity -6. But -6^2 means -(6 · 6) = -36, because the base is only the 6 and the negative sign stays outside. They differ whenever the exponent is even. With an odd exponent, such as (-2)^3 and -2^3, both equal -8 — but do not rely on that; always identify the base first.

What do I do if a variable appears more than once in the expression?

Substitute the same value everywhere it appears. In x^2 - 4x + 1 with x = -3, you write (-3)^2 - 4(-3) + 1 = 9 + 12 + 1 = 22. A variable has one value per problem, so it cannot be -3 in one term and something else in another.

How do I handle fractions and decimals in the same expression?

Convert everything into whichever form is easier for that problem. If the decimals terminate neatly, such as 0.5 = 1/2 or 0.75 = 3/4, fractions usually keep the arithmetic exact. If the fractions are all tenths or hundredths, decimals are faster. Just avoid rounding partway through, since a rounded intermediate value can throw off the final answer.

M7MATH-5.2

Evaluating Expressions

Learn to evaluate one- and two-variable expressions in Grade 7: substitute rational values in parentheses, follow the order of operations, and handle negatives raised to powers.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Evaluating Expressions, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

Start Evaluating Expressions free

What this lesson covers

You already know how to write an expression like

3x - 5

. Now comes the payoff: putting a number in for the variable and finding out what the expression is worth. That process is called evaluating, and it is the bridge between algebra and real answers — the temperature after a drop, the cost of a repair, the value of a formula.

The math itself is just substitution plus the order of operations. The trouble almost always comes from one small habit: dropping the parentheses when you substitute. Once a negative number or a fraction goes in without parentheses,

-4

squared quietly turns into something else entirely. In this lesson you will learn the substitute-in-parentheses rule, work through the order of operations step by step, and get very clear on the difference between

(-4)^2

and

-4^2

Substitution: Always Use Parentheses

To evaluate an expression, you replace each variable with its given value and then simplify to a single number.

The safe way to do this is to write the value inside parentheses every single time. Erase the letter, put empty parentheses where it was, then drop the number in.

Evaluate

5x

when

x = -3

5x \rightarrow 5(\ ) \rightarrow 5(-3) = -15

Why bother? Because algebra hides multiplication. In

5x

, the

5

and the

x

are multiplied even though no symbol shows it. If you substitute without parentheses you get

5-3

, which reads as subtraction and gives

2

— a completely different answer. Parentheses preserve the multiplication.

Parentheses also protect exponents and fractions. In

x^2

with

x = -6

, writing

-6^2

makes the exponent apply only to the

6

; writing

(-6)^2

makes it apply to the whole negative number, which is what the expression means. And in

4x

with

x = \frac{2}{3}

, the form

4\left(\frac{2}{3}\right)

is clearly multiplication, while

4\frac{2}{3}

looks like a mixed number.

One more rule: every appearance of the same letter gets the same value. In

x^2 + 3x

with

x = -2

, you substitute twice:

(-2)^2 + 3(-2)

Expression	Value	Correct substitution	Common wrong start
$7n$	$n = -4$	$7(-4)$	$7-4$
$a^2$	$a = -5$	$(-5)^2$	$-5^2$
$6k$	$k = \frac{1}{2}$	$6\left(\frac{1}{2}\right)$	$6\frac{1}{2}$

Applying the Order of Operations After You Substitute

Once every variable is replaced by a number in parentheses, the problem is pure arithmetic. Follow the order of operations:

Grouping symbols first (parentheses, brackets, and the bar of a fraction), then exponents, then multiplication and division from left to right, then addition and subtraction from left to right.

Evaluate

4 + 3(y - 5)^2

when

y = 2

4 + 3\bigl((2) - 5\bigr)^2

Inside the grouping symbols first:

2 - 5 = -3

, so the expression becomes

4 + 3(-3)^2

.

Exponent next:

(-3)^2 = 9

, giving

4 + 3(9)

.

Multiply:

3 \cdot 9 = 27

, giving

4 + 27 = 31

.

Two cautions. First, multiplication and division are done together, left to right — in

24 \div 3 \cdot 2

you divide first because it comes first, getting

16

, not

4

. The same applies to addition and subtraction.

Second, a fraction bar is a grouping symbol. In

\frac{x + 7}{2}

with

x = -1

, you compute the whole top,

-1 + 7 = 6

, before dividing:

\frac{6}{2} = 3

. Students who divide only the

7

get a wrong result.

Write one step per line and keep the equals signs lined up. Most errors in this lesson are not conceptual — they are two operations crammed into one line, where a sign gets lost.

Negative Numbers Raised to a Power

This is the single biggest source of wrong answers in the lesson, so it deserves its own rule.

An exponent applies only to the thing directly to its left. In

(-4)^2

, the thing to the left of the exponent is the whole quantity

(-4)

, so you multiply

(-4)(-4) = 16

. In

-4^2

, the thing to the left is just the

4

, so you square the

4

and keep the negative in front:

-(4 \cdot 4) = -16

Expression	Meaning	Value
$(-4)^2$	$(-4)(-4)$	$16$
$-4^2$	$-(4 \cdot 4)$	$-16$
$(-2)^3$	$(-2)(-2)(-2)$	$-8$
$-2^3$	$-(2 \cdot 2 \cdot 2)$	$-8$

Notice the last two rows agree. With an odd exponent the two forms happen to match, because the product of an odd number of negatives is negative anyway. With an even exponent they disagree in sign every time. That is why you cannot rely on memory — you have to read what the base actually is.

A useful pattern: a negative base raised to an even power gives a positive result; raised to an odd power it gives a negative result.

Now apply this to an expression. Evaluate

-x^2

when

x = -5

. Substitute in parentheses:

-(-5)^2

. The base is

(-5)

, so

(-5)^2 = 25

, and the leading negative sign makes the answer

-25

. Students who answer

25

usually cancelled the two negative signs before doing the exponent — but exponents come before that sign change.

Two Variables and Rational Values

An expression with two variables works exactly the same way; you just substitute both values before simplifying. Evaluate

3a - 2b

when

a = -1

and

b = 4

3(-1) - 2(4) = -3 - 8 = -11

When the values are fractions or decimals, the arithmetic gets longer but the structure does not change. Evaluate

8m + n

when

m = \frac{3}{4}

and

n = -1.5

8\left(\frac{3}{4}\right) + (-1.5) = 6 + (-1.5) = 4.5

A fraction raised to a power needs care:

\left(\frac{2}{3}\right)^2 = \frac{2}{3} \cdot \frac{2}{3} = \frac{4}{9}

. Square both the numerator and the denominator; squaring only the top is a frequent slip. And

\left(-\frac{1}{2}\right)^3 = -\frac{1}{8}

, negative because the exponent is odd.

It often helps to convert so that all the numbers are in one form. If an expression mixes

0.25

and

\frac{1}{3}

, fractions are usually easier, since

0.25 = \frac{1}{4}

and

\frac{1}{3}

has no clean decimal form.

Finally, adjacent variables mean multiplication. In

xy

with

x = -3

and

y = -6

, substitution gives

(-3)(-6) = 18

. In

5xy

with those same values you get

5(-3)(-6) = 90

. Do not read

xy

as a two-digit number or as

x + y

Where Students Actually Go Wrong

Here are the errors that show up most often on homework and quizzes, with the fix for each.

Dropping parentheses. Substituting

-3

for

x

5x

and writing

5-3

. Fix: replace the variable with empty parentheses first, then fill them in.

Misreading the base of an exponent. Treating

-x^2

as if it were

(-x)^2

. Fix: identify the base — the quantity immediately left of the exponent — before you compute anything.

Doing addition before multiplication. In

2 + 3(4)

, writing

5(4) = 20

instead of

2 + 12 = 14

. Fix: scan for exponents, then multiplication and division, then addition and subtraction.

Ignoring the fraction bar as grouping. In

\frac{2x + 6}{4}

with

x = 1

, computing

2(1) + \frac{6}{4}

. Fix: put invisible parentheses around the whole numerator and the whole denominator.

Subtracting a negative incorrectly. In

7 - y

with

y = -2

, writing

7 - 2 = 5

. Substituting properly gives

7 - (-2) = 7 + 2 = 9

.

Sign drift. Copying

-27

27

on the next line. Fix: one operation per line, and reread the sign before you rewrite it.

One last habit worth building: after you finish, ask whether the answer is reasonable. If every term is negative, the total cannot be positive. If you multiplied a number by a proper fraction, the result should be smaller than the number you started with. This estimate check catches sign errors quickly, and it becomes essential later when you evaluate formulas for area, temperature, and rates of change.

Key terms

Evaluate.: To replace each variable in an expression with a given value and simplify the result to a single number.
Substitute.: To put a given numerical value in place of a variable, written inside parentheses to preserve multiplication and grouping.
Variable.: A letter that stands for a number whose value can change or is given in the problem.
Base.: The quantity an exponent applies to — the number or parenthesized expression immediately to the left of the exponent.
Exponent.: A small raised number showing how many times the base is used as a factor, as in $(-3)^4 = (-3)(-3)(-3)(-3)$ .
Order of operations.: The agreed sequence for simplifying: grouping symbols, then exponents, then multiplication and division left to right, then addition and subtraction left to right.
Rational value.: A number that can be written as a ratio of two integers, including integers, fractions, and terminating or repeating decimals — positive or negative.
Coefficient.: The number multiplied by a variable, such as the $5$ in $5x$ ; substitution turns $5x$ into $5$ times the given value.

Worked example

Evaluate

2x^2 - 3xy + 4

when

x = -3

and

y = \frac{1}{2}

Step 1 — Rewrite with empty parentheses. Mark where each variable goes so nothing runs together:

2(\ )^2 - 3(\ )(\ ) + 4

.

Step 2 — Substitute. Put

-3

in for each

x

and

\frac{1}{2}

in for

y

2(-3)^2 - 3(-3)\left(\frac{1}{2}\right) + 4

Step 3 — Exponents first. The base is

(-3)

, not

3

, so

(-3)^2 = (-3)(-3) = 9

. The expression becomes

2(9) - 3(-3)\left(\frac{1}{2}\right) + 4

Step 4 — Multiplication, left to right. First term:

2(9) = 18

. Second term:

3(-3) = -9

, then

-9 \cdot \frac{1}{2} = -4.5

. Because the expression subtracts that product, the term contributes

-(-4.5) = +4.5

. Written out:

18 + 4.5 + 4

Step 5 — Add left to right.

18 + 4.5 = 22.5

, then

22.5 + 4 = 26.5

.

Answer: $26.5$ , which is also

\frac{53}{2}

.

Check the signs. The middle term was subtraction of a negative product, so it had to increase the total. Since

18 + 4 = 22

already, an answer above

22

is reasonable. A student who wrote

2(-3)^2

-18

would get

-14.5

— the tell-tale sign of missing the parentheses around the base.

Practice questions

What is the value of

-n^2

when

n = -5

$25$
$-25$
$-10$
$10$

Answer: $-25$

Substitute in parentheses:

-(-5)^2

. Exponents come before the leading negative sign, so first compute

(-5)^2 = (-5)(-5) = 25

. The negative sign in front of the expression then makes the result

-25

. Answering

25

means the two negatives were combined before the exponent was applied; answering

-10

means the exponent was treated as multiplication by

2

Evaluate

5 - 2(m - n)^3

when

m = -1

and

n = 2

. Show each step.

Answer: $59$

Substitute:

5 - 2\bigl((-1) - (2)\bigr)^3

. Grouping symbols first:

-1 - 2 = -3

, giving

5 - 2(-3)^3

. Exponent next:

(-3)^3 = (-3)(-3)(-3) = -27

, an odd power of a negative, so the result is negative. Now

5 - 2(-27)

. Multiply before subtracting:

2(-27) = -54

, so the expression is

5 - (-54) = 5 + 54 = 59

. The most common slip is subtracting

54

to get

-49

, which happens when the subtraction sign and the negative product are not tracked separately.

Nadia evaluated

4 + 3x

for

x = -2

and got

-6

. Explain her mistake and give the correct value.

Answer: She added before multiplying (treating it as $7 \cdot (-2)$ ); the correct value is $-2$ .

Substituting gives

4 + 3(-2)

. The order of operations requires the multiplication

3(-2) = -6

before any addition, so the expression is

4 + (-6) = -2

. Nadia combined

4 + 3

to get

7

and then multiplied by

-2

, producing

-14

— or, in her version, she mishandled the signs and landed on

-6

, which is just the product with the

4

left out. Either way, the fix is the same: multiplication happens before addition, and the

4

stays as its own term.

FAQ

Why do I have to put the substituted number in parentheses?: Because algebra hides multiplication signs and exponent bases. In $6x$ the multiplication is invisible, so substituting $-2$ without parentheses gives $6-2$ , which reads as subtraction. And in $x^2$ , writing $-2^2$ makes the exponent apply only to the $2$ . Parentheses keep the meaning of the original expression intact, so they cost you nothing and prevent the two most common errors.
Is $(-6)^2$ the same as $-6^2$ ?: No. $(-6)^2$ means $(-6)(-6) = 36$ , because the base is the whole quantity $-6$ . But $-6^2$ means $-(6 \cdot 6) = -36$ , because the base is only the $6$ and the negative sign stays outside. They differ whenever the exponent is even. With an odd exponent, such as $(-2)^3$ and $-2^3$ , both equal $-8$ — but do not rely on that; always identify the base first.
What do I do if a variable appears more than once in the expression?: Substitute the same value everywhere it appears. In $x^2 - 4x + 1$ with $x = -3$ , you write $(-3)^2 - 4(-3) + 1 = 9 + 12 + 1 = 22$ . A variable has one value per problem, so it cannot be $-3$ in one term and something else in another.
How do I handle fractions and decimals in the same expression?: Convert everything into whichever form is easier for that problem. If the decimals terminate neatly, such as $0.5 = \frac{1}{2}$ or $0.75 = \frac{3}{4}$ , fractions usually keep the arithmetic exact. If the fractions are all tenths or hundredths, decimals are faster. Just avoid rounding partway through, since a rounded intermediate value can throw off the final answer.

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