Evaluate and Identify Equivalent Exponent Expressions | Grade 8 Math | LevelKite

In this lesson you use the four properties of exponents you already know to work with expressions that contain powers. A power such as $2^5$ has a base and an exponent. The base, 2, is the repeated factor, and the exponent, 5, counts how many times it appears: $2^5 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32$ .

You have two goals:

Evaluate expressions. To evaluate an expression means to find the single number it stands for.
Identify equivalent expressions. Expressions are equivalent when they have the same value, even though they are written differently.

The table below lists those four properties. In each rule, $a$ and $b$ stand for numbers, and $m$ and $n$ stand for positive whole numbers. When a base is negative, write it in parentheses, as in $(-2)^3$ .

Each rule is true only under its conditions. A base in a denominator cannot be $0$ , because division by $0$ is undefined. The quotient of powers also needs $m > n$ , so that the new exponent $m - n$ is a positive whole number: it counts the factors left on top after the factors on the bottom divide out.

Four properties of exponents you already know

Property	Rule	Conditions	Example
Product of powers	$a^m \cdot a^n = a^{m+n}$	Same base $a$	$5^2 \cdot 5^3 = 5^5 = 3125$
Quotient of powers	$\frac{a^m}{a^n} = a^{m-n}$	Same base $a$ , with $a \neq 0$ and $m > n$	$\frac{7^6}{7^4} = 7^2 = 49$
Power of a product	$(ab)^n = a^n b^n$	Any numbers $a$ and $b$	$(2 \cdot 5)^3 = 2^3 \cdot 5^3 = 8 \cdot 125 = 1000$
Power of a quotient	$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$	$b \neq 0$	$\left(\frac{3}{4}\right)^2 = \frac{3^2}{4^2} = \frac{9}{16}$

Key concept

Rewrite with properties, then evaluate or compare

Each property of exponents is an equation that is true whenever its conditions hold. So when you rewrite an expression with a property, the new expression is equivalent to the old one: it has the same value. Link your rewrites with equals signs in one chain. To evaluate, follow the chain to a single power or a short product, then compute. To decide whether two expressions are equivalent, simplify each one and then compare them.

Worked example

Evaluate $\dfrac{2^5 \cdot 2^4}{2^6}$ .

The top is a product of two powers with the same base, 2, so use the product of powers and add the exponents. Five factors of 2 followed by four more make nine factors of 2.
$\dfrac{2^5 \cdot 2^4}{2^6} = \dfrac{2^{5+4}}{2^6} = \dfrac{2^9}{2^6}$
Now use the quotient of powers. It applies because both powers have base 2, the base is not 0, and the top exponent is larger (9 > 6). Subtract the exponents: the six factors of 2 on the bottom divide out with six factors on top.
$\dfrac{2^9}{2^6} = 2^{9-6} = 2^3$
Evaluate the single power by multiplying its three factors of 2.
$2^3 = 2 \cdot 2 \cdot 2 = 8$
Check by evaluating each power first. This route needs more arithmetic, and its chain also ends at 8.
$\dfrac{2^5 \cdot 2^4}{2^6} = \dfrac{32 \cdot 16}{64} = \dfrac{512}{64} = 8$

Answer

The value of $\dfrac{2^5 \cdot 2^4}{2^6}$ is $8$ .

Worked example

Evaluate $\left(\dfrac{6}{5}\right)^3 \cdot \dfrac{5^5}{2^3}$ .

The first factor is a power of a quotient. Its denominator, 5, is not 0, so raise the top and the bottom each to the third power.
$\left(\dfrac{6}{5}\right)^3 \cdot \dfrac{5^5}{2^3} = \dfrac{6^3}{5^3} \cdot \dfrac{5^5}{2^3}$
Multiply the fractions by multiplying the tops and multiplying the bottoms. The bottom factors may go in either order, so put the power of 2 first. Then split the result into two fractions, each pairing powers you can simplify.
$\dfrac{6^3}{5^3} \cdot \dfrac{5^5}{2^3} = \dfrac{6^3 \cdot 5^5}{2^3 \cdot 5^3} = \dfrac{6^3}{2^3} \cdot \dfrac{5^5}{5^3}$
A property is an equation, so it also works from right to left. The first fraction has the same exponent, 3, on top and bottom, and 2 is not 0, so the power of a quotient combines it into one power of 6/2, which is 3. The second fraction has base 5, which is not 0, and a larger top exponent, so subtract the exponents.
$\dfrac{6^3}{2^3} \cdot \dfrac{5^5}{5^3} = \left(\dfrac{6}{2}\right)^3 \cdot 5^{5-3} = 3^3 \cdot 5^2$
Evaluate each power, then multiply.
$3^3 \cdot 5^2 = 27 \cdot 25 = 675$

Answer

The value of $\left(\dfrac{6}{5}\right)^3 \cdot \dfrac{5^5}{2^3}$ is $675$ .

Now for the second goal: deciding whether two expressions are equivalent. One way is to evaluate both and compare the numbers. Another way is to use the properties to rewrite both as single powers of the same base and then compare the exponents.

Comparing exponents uses one new rule, and the rule works only when the base $b$ satisfies $b > 0$ and $b \neq 1$ . Here is why. If $b > 1$ , each extra factor of $b$ makes a power larger. If $0 < b < 1$ , each extra factor makes a power smaller. Either way, different exponents give different values. Without these conditions, equal values can hide different exponents: $1^2 = 1 = 1^5$ and $(-1)^2 = 1 = (-1)^4$ .

Here is the rule, where $x$ and $y$ are positive whole numbers:

b^x = b^y \text{ exactly when } x = y, \quad \text{for } b > 0 \text{ and } b \neq 1

Worked example

Which of these expressions are equivalent to $2^6$ : (A) $4^3$ , (B) $2^2 \cdot 2^3$ , (C) $(-2)^6$ , (D) $-2^6$ ?

First find the value of 2 to the sixth power, the expression to match. Then rewrite each choice as a power of 2 or as a number. Because the base 2 is positive and not 1, a power of 2 with a positive whole-number exponent equals 2 to the sixth power only when that exponent is 6.
$2^6 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 64$
(A) Write 4 as 2 times 2. Apply the power of a product, then the product of powers, since both factors have base 2. The chain ends at 2 to the sixth power, so (A) is equivalent to it.
$4^3 = (2 \cdot 2)^3 = 2^3 \cdot 2^3 = 2^{3+3} = 2^6$
(B) The product of powers gives a power of 2 with exponent 5. Since 5 is not 6 and the base 2 is positive and not 1, the values differ. The numbers confirm it: 32 is not 64. So (B) is not equivalent.
$2^2 \cdot 2^3 = 2^{2+3} = 2^5 = 32 \neq 64$
(C) The parentheses make -2 the base. Write -2 as -1 times 2 and apply the power of a product. The six factors of -1 form three pairs, and each pair multiplies to 1, so (-1) to the sixth power is 1. The chain ends at 2 to the sixth power, so (C) is equivalent.
$(-2)^6 = (-1 \cdot 2)^6 = (-1)^6 \cdot 2^6 = 1 \cdot 2^6 = 2^6$
(D) With no parentheses, the base is 2, so find 2 to the sixth power first. The minus sign then takes its opposite. A negative number cannot equal 64, so (D) is not equivalent.
$-2^6 = -(2^6) = -64 \neq 64$

Answer

Choices (A) $4^3$ and (C) $(-2)^6$ are equivalent to $2^6$ , while choices (B) $2^2 \cdot 2^3$ and (D) $-2^6$ are not.

Takeaway: Each property of exponents turns an expression into an equivalent one, as long as its conditions hold. To evaluate, link rewrites with equals signs until you reach a simple form, then compute. To decide whether two expressions are equivalent, simplify both and compare their values, or compare the exponents of single powers with the same base $b$ , where $b > 0$ and $b \neq 1$ . Read every base carefully: in $(-a)^n$ the base is $-a$ , but in $-a^n$ the base is $a$ .