For all x > 0, the expression \frac{\sqrt{x^7}}{\sqrt[5]{x^3}} is equivalent to x^a, where… — SAT

For all $x > 0$ , the expression $\frac{\sqrt{x^7}}{\sqrt[5]{x^3}}$ is equivalent to $x^a$ , where $a$ is a constant. What is the value of $a$ ?

Answer: 2.9

2.9

To rewrite the expression, we convert the radicals to rational exponents:

\sqrt{x^7} = x^{7/2}

and

\sqrt[5]{x^3} = x^{3/5}

. Using the quotient property of exponents,

\frac{x^m}{x^n} = x^{m-n}

, we subtract the denominator's exponent from the numerator's exponent:

\frac{7}{2} - \frac{3}{5}

. Finding a common denominator of 10 gives

\frac{35}{10} - \frac{6}{10} = \frac{29}{10}

, which simplifies to

2.9

. Thus, the value of

a

2.9

Convert the radical expressions to rational exponents.

Numerator:

\sqrt{x^7} = x^{7/2}

; Denominator:

\sqrt[5]{x^3} = x^{3/5}

Radical expressions

\sqrt[n]{x^m}

can be rewritten as rational exponent expressions

x^{m/n}

Apply the quotient rule of exponents.

x^{7/2} / x^{3/5} = x^{7/2 - 3/5}

When dividing exponential expressions with the same base, subtract the exponent in the denominator from the exponent in the numerator.

Subtract the exponents by finding a common denominator.

\frac{7}{2} - \frac{3}{5} = \frac{35}{10} - \frac{6}{10} = \frac{29}{10} = 2.9

Subtracting the fractions requires a common denominator of 10. The result

\frac{29}{10}

is equivalent to the terminating decimal

2.9

Equivalent algebraic expressions involving radical and rational exponents

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