Same-Base Products, Quotients and Powers of Powers | Grade 8 Math | LevelKite

Recall that an exponent counts repeated factors. In $2^4 = 2 \cdot 2 \cdot 2 \cdot 2 = 16$ , the base $2$ is the repeated factor, the exponent $4$ counts the factors, and $2^4$ is a power of 2. Because an exponent is a count, you can combine powers by counting factors instead of multiplying everything out.

In this lesson every exponent is a positive whole number. You will multiply and divide powers with the same base, then raise a power to a power. Each time, you will write an equivalent expression: one with the same value as the original.

Multiplying powers. Write $2^3 \cdot 2^4$ out as factors, as shown below. The first power gives three factors of 2 and the second gives four more, so the product has $3 + 4 = 7$ factors of 2. Check: $2^3 \cdot 2^4 = 8 \cdot 16 = 128$ , and $2^7 = 128$ . So to multiply powers with the same base, keep the base and add the exponents.

2^3 \cdot 2^4 = (2 \cdot 2 \cdot 2) \cdot (2 \cdot 2 \cdot 2 \cdot 2) = 2^{3+4} = 2^7

Dividing powers. Write $3^5 \div 3^2$ as a fraction and expand both powers, as shown below. Each factor of 3 on the bottom pairs with a factor of 3 on top, and each pair $\frac{3}{3}$ equals $1$ . The two pairs use up two of the five factors on top, so $5 - 2 = 3$ factors of 3 remain. Check: $3^5 \div 3^2 = 243 \div 9 = 27$ , and $3^3 = 27$ .

The pairing needs a nonzero base. With a base of $0$ , the bottom would be $0$ , and division by $0$ is undefined. So to divide powers with the same nonzero base, keep the base and subtract the bottom exponent from the top exponent. In this lesson the top exponent is always larger, so some factors are left over.

\frac{3^5}{3^2} = \frac{3 \cdot 3 \cdot 3 \cdot 3 \cdot 3}{3 \cdot 3} = \frac{3}{3} \cdot \frac{3}{3} \cdot (3 \cdot 3 \cdot 3) = 1 \cdot 1 \cdot 3^{5-2} = 3^3

Raising a power to a power. A power of a power is a power raised to another exponent, such as $(4^2)^3$ . The exponent outside the parentheses, 3, says to use $4^2$ as a factor three times, and each $4^2$ holds two factors of 4, as shown below. Three groups of two factors make $3 \cdot 2 = 6$ factors of 4. Check: $4^2 = 16$ , and $16 \cdot 16 \cdot 16 = 256 \cdot 16 = 4096$ , which matches $4^6 = 4096$ . So to raise a power to a power, keep the base and multiply the exponents.

(4^2)^3 = 4^2 \cdot 4^2 \cdot 4^2 = (4 \cdot 4) \cdot (4 \cdot 4) \cdot (4 \cdot 4) = 4^{3 \cdot 2} = 4^6

Key concept

Combine same-base powers by counting factors

An exponent counts factors of its base, so powers with the same base combine by counting factors: add the exponents to multiply, subtract them to divide, and multiply them to raise a power to a power.

In these rules $a$ is any number, and $m$ and $n$ are positive whole numbers. To divide, $a$ must not be zero, and in this lesson $m > n$ . Write a negative base in parentheses: $(-2)^4 = 16$ has base $-2$ , but $-2^4 = -(2^4) = -16$ has base $2$ .

a^m \cdot a^n = a^{m+n} \qquad \frac{a^m}{a^n} = a^{m-n} \ (a \neq 0,\ m > n) \qquad (a^m)^n = a^{mn}

Worked example

Multiply, then divide

Write $\dfrac{6^4 \cdot 6^5}{6^7}$ as a single power of 6, then find its value.

Multiply the powers on top. They share the base 6, so the 4 factors of 6 join 5 more, making $4 + 5 = 9$ factors. Add the exponents.
$6^4 \cdot 6^5 = 6^{4+5} = 6^9$
Divide. The base 6 is not zero, and the top exponent 9 is larger than the bottom exponent 7. Each of the seven factors of 6 on the bottom pairs with one on top to make $\frac{6}{6} = 1$ , so $9 - 7 = 2$ factors remain. Subtract the exponents.
$\frac{6^9}{6^7} = 6^{9-7} = 6^2$
Evaluate the single power: $6^2$ means two factors of 6.
$6^2 = 6 \cdot 6 = 36$

Answer

The expression equals $6^2$ , which is $36$ .

Worked example

A negative base

Write $\dfrac{(-2)^2 \cdot (-2)^5}{(-2)^4}$ as a single power of $-2$ , then find its value.

Read the base first, because the product and quotient rules need powers of the same base. Each power has $-2$ inside parentheses, so the base is $-2$ every time.
Multiply on top. The base is the same, so 2 factors of $-2$ join 5 more. Add the exponents.
$(-2)^2 \cdot (-2)^5 = (-2)^{2+5} = (-2)^7$
Divide. The base $-2$ is not zero, so each pair $\frac{-2}{-2}$ equals $1$ , and $7 > 4$ , so factors remain. Subtract the exponents.
$\frac{(-2)^7}{(-2)^4} = (-2)^{7-4} = (-2)^3$
Evaluate. Two factors of $-2$ multiply to $4$ , and the third factor of $-2$ makes the product negative.
$(-2)^3 = (-2)(-2)(-2) = 4 \cdot (-2) = -8$
Check with the original values: $(-2)^2 = 4$ , $(-2)^5 = -32$ and $(-2)^4 = 16$ .
$\frac{4 \cdot (-32)}{16} = \frac{-128}{16} = -8$

Answer

The expression equals $(-2)^3$ , which is $-8$ .

Worked example

Choosing the right rule

Write $\dfrac{(3^2)^4}{3^4 \cdot 3^2}$ as a single power of 3, then find its value.

The top is a power of a power: four copies of $3^2$ hold $4 \cdot 2 = 8$ factors of 3. Multiply the exponents.
$(3^2)^4 = 3^{2 \cdot 4} = 3^8$
The bottom is a product with the same base: a group of 4 factors of 3 joins a group of 2. Add the exponents. The same numbers, 4 and 2, give 8 factors on top but 6 here, because four copies of 2 factors make $4 \cdot 2 = 8$ , while 4 factors joined with 2 more make $4 + 2 = 6$ .
$3^4 \cdot 3^2 = 3^{4+2} = 3^6$
Divide. The base 3 is not zero and $8 > 6$ , so subtract the exponents, then evaluate.
$\frac{3^8}{3^6} = 3^{8-6} = 3^2 = 9$
Check with values. The top is $(3^2)^4 = 9^4 = 81 \cdot 81 = 6561$ , and the bottom is $3^4 \cdot 3^2 = 81 \cdot 9 = 729$ . Since $729 \cdot 9 = 6561$ , the quotient is 9.
$\frac{6561}{729} = 9$

Answer

The expression equals $3^2$ , which is $9$ .

Takeaway: An exponent counts factors of its base, so powers with the same base combine by counting. Add the exponents to multiply, subtract them to divide (with a nonzero base and a larger top exponent), and multiply them to raise a power to a power. Next, you will use the same counting to find the power of a product and the power of a quotient.