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Use the Quotient Property and Negative Exponent Property to generate an equivalent expression for √49/7^-3. What will the rational exponent be for the new expression?(1 point)

7^__

3 answers

To simplify the expression \(\sqrt{\frac{49}{7^{-3}}}\), we'll begin by rewriting it using the properties of exponents.

  1. Rewrite the square root as a rational exponent: \[ \sqrt{\frac{49}{7^{-3}}} = \left(\frac{49}{7^{-3}}\right)^{\frac{1}{2}} \]

  2. Use the Quotient Property of exponents: \[ \frac{49}{7^{-3}} = 49 \cdot 7^3 \] Here, we used the property \(\frac{a}{b^{-c}} = a \cdot b^c\).

  3. Now simplify \(49\) as a power of \(7\): \[ 49 = 7^2 \] Therefore, \[ 49 \cdot 7^3 = 7^2 \cdot 7^3 \]

  4. Use the Product Property of exponents: \[ 7^2 \cdot 7^3 = 7^{2+3} = 7^5 \]

  5. Now substitute back into the expression with the rational exponent: \[ \left(\frac{49}{7^{-3}}\right)^{\frac{1}{2}} = (7^5)^{\frac{1}{2}} \]

  6. Finally, use the Power Property of exponents: \[ (7^5)^{\frac{1}{2}} = 7^{5 \cdot \frac{1}{2}} = 7^{\frac{5}{2}} \]

Therefore, the equivalent expression for \(\sqrt{\frac{49}{7^{-3}}}\) is \(7^{\frac{5}{2}}\).

The rational exponent for the new expression is \(\frac{5}{2}\).

Final answer: \[ 7^{\frac{5}{2}} \]

try again

Let's simplify the expression \(\sqrt{\frac{49}{7^{-3}}}\) step-by-step, using the Quotient Property and Negative Exponent Property correctly.

  1. Start with the initial expression: \[ \sqrt{\frac{49}{7^{-3}}} \]

  2. Rewrite the square root: \[ \sqrt{\frac{49}{7^{-3}}} = \left(\frac{49}{7^{-3}}\right)^{1/2} \]

  3. Simplify the fraction \(\frac{49}{7^{-3}}\): Using the property \(a^{-b} = \frac{1}{a^b}\): \[ \frac{49}{7^{-3}} = 49 \cdot 7^3 \]

  4. Rewrite \(49\) as \(7^2\): \[ 49 = 7^2 \] Therefore, \[ 49 \cdot 7^3 = 7^2 \cdot 7^3 \]

  5. Combine the exponents: \[ 7^2 \cdot 7^3 = 7^{2 + 3} = 7^5 \]

  6. Substitute this back into the square root: \[ \left(7^5\right)^{1/2} \]

  7. Use the Power Property of exponents: \[ (7^5)^{1/2} = 7^{5 \cdot \frac{1}{2}} = 7^{\frac{5}{2}} \]

Thus, the final expression for \(\sqrt{\frac{49}{7^{-3}}}\) simplifies to \(7^{\frac{5}{2}}\).

So the rational exponent is: \[ \frac{5}{2} \]

Final answer: 7^{\frac{5}{2}}