Question

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simplify the following expressions using the correct laws of exponents. ensure each expression is reduced to its simplest form.

1. $2^3 \cdot 2^4 = \square$
2. $\frac{8^5}{8^3} = \square$
3. $(3^2)^3 = \square$
4. $\sqrt3{x^9} \square$
5. $5^0 = \square$
6. $(mn)^3 = \square$
7. $10^{-3} = \square$

Explanation:

1. Simplify \(2^{3} \cdot 2^{4}\)

Step1: Use product of exponents rule (\(a^m \cdot a^n = a^{m + n}\))

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

Step2: Calculate the exponent

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

Step3: Evaluate \(2^7\)

\(2^{7}=128\)

Step1: Use quotient of exponents rule (\(\frac{a^m}{a^n}=a^{m - n}\))

\(\frac{8^{5}}{8^{3}} = 8^{5 - 3}\)

Step2: Calculate the exponent

\(8^{5 - 3}=8^{2}\)

Step3: Evaluate \(8^2\)

\(8^{2}=64\)

Step1: Use power of a power rule (\((a^m)^n=a^{m\times n}\))

\((3^{2})^{3}=3^{2\times3}\)

Step2: Calculate the exponent

\(3^{2\times3}=3^{6}\)

Step3: Evaluate \(3^6\)

\(3^{6} = 729\)

Answer:

\(128\)

2. Simplify \(\frac{8^{5}}{8^{3}}\)