Laws of Exponents: Practice

Lining up the factors of one power next to the factors of the other gives a single run of factors whose length is the sum of the counts.

$a^m \times a^n = a^{m+n}$

So you add the exponents and keep the base.

Same base, and the top exponent is larger, so subtract the exponents.

$\frac{3^6}{3^2} = 3^{6-2} = 3^4$

Six factors of $3$ with two cancelled leaves four factors.

Same base, larger exponent on top, so subtract.

$\frac{8^5}{8^3} = 8^{5-3} = 8^2$

A power raised to a power multiplies the exponents.

$(2^3)^2 = 2^{3 \times 2} = 2^6$

Two groups of three factors is six factors in all.

Same base, so add the exponents.

$4^2 \times 4^2 = 4^{2+2} = 4^4$

Raise a power to a power by multiplying the exponents.

$(5^2)^3 = 5^{2 \times 3} = 5^6$

A power of a product puts the exponent on each factor separately.

$(2 \times 3)^2 = 2^2 \times 3^2$

Both sides equal $36$: $6^2 = 36$ and $4 \times 9 = 36$.

Same base, larger exponent on top, so subtract.

$\frac{6^7}{6^4} = 6^{7-4} = 6^3$

Multiply the exponents; multiplying by $1$ changes nothing.

$(3^4)^1 = 3^{4 \times 1} = 3^4$

Same base, so add the exponents.

$2^5 \times 2^1 = 2^{5+1} = 2^6$

Same base, larger exponent on top, so subtract.

$\frac{4^9}{4^8} = 4^{9-8} = 4^1$

This is also just $4$.

Raise a power to a power by multiplying the exponents.

$(2^4)^2 = 2^{4 \times 2} = 2^8$