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Radical Expressions and Equations

Combine your skills of working with exponents and manipulating expressions for radical glory.

Warmup

         

If \( 3 ^ N = 3 ^ 5 \times 3 ^ 4\), what is the value of \(N\)?

What value of \(N\) satisfies the equation

\[ \frac { 8 ^ {22}} { 8 ^ {11}} = 8 ^ N? \]

Given \(3^N = 9 ^{10} \), what is the value of \(N\)?

Hint: Rewrite 9 as \(3^2.\)

If \(x\) is a real number satisfying \( 14 ^x = 3 \), what is the value of \( 14 ^{2x + 1} ?\)

\(x\) and \(y\) satisfy \(2^x = 10\) and \(10 ^ y = 128\). What is the value of \(xy\)?

Hint: Rewrite the "10" in \(10^y = 128\) by using the fact that \(2^x = 10.\)

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