Question

1. use power laws to simplify the following expressions then evaluate.

a) \\((2^6)(2^{-3})\\)
b) \\((3^3)_2\\)
c) \\(\frac{7^{11}}{(7^2 \times 7^3)^3}\\)
d) \\((-8)^0\\)

Explanation:

Rotate the image mentally to read the expressions

The image is upside down. Let's read the text and expressions by rotating them 180 degrees:
"3. Use power laws to simplify the following expressions then evaluate."
The sub-questions are:
a) \((2^6)(2^{-3})\)
b) \((3^3)^2\)
c) \(\frac{(7^2 \times 7^3)^3}{7^{11}}\)
d) \((-8)^0\)

Simplify and evaluate expression a

Apply the product of powers rule: \(a^m \cdot a^n = a^{m+n}\).

$$ (2^6)(2^{-3}) = 2^{6 + (-3)} = 2^3 = 8 $$

Simplify and evaluate expression b

Apply the power of a power rule: \((a^m)^n = a^{m \cdot n}\).

$$ (3^3)^2 = 3^{3 \cdot 2} = 3^6 = 729 $$

Simplify and evaluate expression c

Apply the product of powers rule inside the parentheses, then the power of a power rule, and finally the quotient of powers rule: \(\frac{a^m}{a^n} = a^{m-n}\).

$$ \frac{(7^2 \times 7^3)^3}{7^{11}} = \frac{(7^{2+3})^3}{7^{11}} = \frac{(7^5)^3}{7^{11}} = \frac{7^{15}}{7^{11}} = 7^{15-11} = 7^4 = 2401 $$

Simplify and evaluate expression d

Apply the zero exponent rule: \(a^0 = 1\) for any non-zero \(a\).

$$ (-8)^0 = 1 $$

Answer:

Question 3a

\(2^3 = 8\)

Question 3b

\(3^6 = 729\)

Question 3c

\(7^4 = 2401\)

Question 3d

\(1\)