Select the equivalent expression. root(10)((1)/(t^(8)*t^(6))) t^(-(7)/(5)) t^((5)/(7)) t^(-(5)/(7)) t^((7)/(5))

Understand given expression: We are given the expression: root(10)((1)/(t^(8)*t^(6))) First, we need to understand that the 10th root of a number can be written as a fractional exponent with the denominator being 10. So, we can rewrite the expression as: (1/(t^(8)*t^(6)))^(1/10)Apply exponent rule: Next, we apply the exponent rule for multiplying with the same base, which states that when we multiply two exponents with the same base, we add the exponents: t^(8)*t^(6) = t^(8+6) = t^(14) So, the expression becomes: (1/t^(14))^(1/10)Apply power of a quotient rule: Now, we apply the power of a quotient rule, which states that (a/b)^n = a^n / b^n. In this case, we have: (1^(1/10)) / (t^(14*(1/10))) Since 1 raised to any power is 1, the numerator remains 1: 1 / t^(14/10)Simplify the exponent: We simplify the exponent by dividing 14 by 10: 14/10 = 7/5 So, the expression simplifies to: 1 / t^(7/5)Write with negative exponent: Finally, we can write the expression with a negative exponent to move the t term to the numerator: t^(-(7/5)) This matches one of the answer choices.

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Select the equivalent expression.

root(10)((1)/(t^(8)*t^(6)))

t^(-(7)/(5))

t^((5)/(7))

t^(-(5)/(7))

t^((7)/(5))

Select the equivalent expression. $\newline$ $\sqrt[10]{\frac{1}{t^{8} \cdot t^{6}}}$ $\newline$ $t^{-\frac{7}{5}}$ $\newline$ $t^{\frac{5}{7}}$ $\newline$ $t^{-\frac{5}{7}}$ $\newline$ $t^{\frac{7}{5}}$

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Math Problems

Algebra 1

Multiplication with rational exponents

Full solution

Q. Select the equivalent expression.

\newline

\sqrt[10]{\frac{1}{t^{8} \cdot t^{6}}}

\newline

t^{-\frac{7}{5}}

\newline

t^{\frac{5}{7}}

\newline

t^{-\frac{5}{7}}

\newline

t^{\frac{7}{5}}

Understand given expression: We are given the expression: $\newline$ $\sqrt[10]{\frac{1}{t^{8}\cdot t^{6}}}$ $\newline$ First, we need to understand that the $10$ th root of a number can be written as a fractional exponent with the denominator being $10$ . $\newline$ So, we can rewrite the expression as: $\newline$ $\left(\frac{1}{t^{8}\cdot t^{6}}\right)^{\frac{1}{10}}$
Apply exponent rule: Next, we apply the exponent rule for multiplying with the same base, which states that when we multiply two exponents with the same base, we add the exponents: $\newline$ $t^{8}$ $\cdot$ $t^{6} = t^{8+6} = t^{14}$ $\newline$ So, the expression becomes: $\newline$ $\left(\frac{1}{t^{14}}\right)^{\frac{1}{10}}$
Apply power of a quotient rule: Now, we apply the power of a quotient rule, which states that $(a/b)^n = a^n / b^n$ . In this case, we have: $\newline$ $(1^{1/10}) / (t^{14*(1/10)})$ $\newline$ Since $1$ raised to any power is $1$ , the numerator remains $1$ : $\newline$ $1 / t^{14/10}$
Simplify the exponent: We simplify the exponent by dividing $14$ by $10$ : $\newline$ $\frac{14}{10} = \frac{7}{5}$ $\newline$ So, the expression simplifies to: $\newline$ $1 / t^{\frac{7}{5}}$
Write with negative exponent: Finally, we can write the expression with a negative exponent to move the $t$ term to the numerator: $t^{-(\frac{7}{5})}$ This matches one of the answer choices.