Question

1. write as a single power. express answers with positive exponents.

a) \\(\frac{a^5}{a^3}\\)
d) \\((d^6)^3\\)
b) \\((b)(b^4)(b^2)\\)
e) \\(\frac{a(c^5)}{c^2}\\)
c) \\(\frac{c^5}{c^9}\\)
f) \\((f^{-3})^{-2}\\)
lesson 7.4

Explanation:

Part (a)

Step1: Apply quotient rule of exponents

When dividing powers with the same base, subtract the exponents: $\frac{a^{m}}{a^{n}} = a^{m - n}$. Here, $m = 5$ and $n = 3$.
$\frac{a^{5}}{a^{3}}=a^{5 - 3}$

Step2: Simplify the exponent

$5-3 = 2$, so $a^{5 - 3}=a^{2}$

Step1: Apply product rule of exponents

When multiplying powers with the same base, add the exponents: $a^{m}\cdot a^{n}=a^{m + n}$. For $(b)(b^{4})(b^{2})$, the base is $b$, and the exponents are $1$, $4$, and $2$ (since $b = b^{1}$).
$(b)(b^{4})(b^{2})=b^{1 + 4+ 2}$

Step2: Simplify the exponent

$1 + 4+ 2=7$, so $b^{1 + 4 + 2}=b^{7}$

Step1: Apply quotient rule of exponents

Using $\frac{a^{m}}{a^{n}}=a^{m - n}$ with base $c$, $m = 5$ and $n = 9$.
$\frac{c^{5}}{c^{9}}=c^{5 - 9}$

Step2: Simplify the exponent and make it positive

$5-9=-4$, and using the rule $a^{-n}=\frac{1}{a^{n}}$, we get $c^{-4}=\frac{1}{c^{4}}$

Answer:

$a^{2}$

Part (b)