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# Common Core Standard HSN-RN.A.1 Questions

Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.

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Which of the following correctly explains why $(root(6){10})^6 = 10$ ? Choose all correct answers.
1. The 6's cancel out, leaving just 10.
2. $root(6){10}$ can be rewritten as $10^{1/6}$, and then exponent laws can be applied (namely, $(10^{1/6})^6 = 10^{6/6} = 10^{1}=10$).
3. If the left hand side is rewritten as $root(6){10^{6}}$, then the radical can be distributed to the 10 and the 6, and the rounded answer is about 10.
4. The 6th root of 10 is asking for the number that, if multiplied by itself 6 times, would equal 10. Therefore, this number raised to the power of 6, is equal to 10.
Starting with the variable x, raise x to the power of 4. Which of the following represents an inverse operation? Choose all correct answers.
1. Dividing the expression by 4.
2. Taking the base-4 logarithm of the expression.
3. Taking the 4th root.
4. Raising the entire expression to the power of 1/4.
Which of the following is a correct way to exactly rewrite the expression $2^{5/7}$? Choose all that apply.
1. $1.6$
2. $root(7)(2^5)$
3. $(root(7)(2))^5$
4. $2^(7/5)$
Which of the following is a correct way to rewrite the expression $4^(-5/3)$ ? Choose all that apply.
1. $root(3)(4^(-5))$
2. $-root(3)(4^5)$
3. $1 / (root(3)(4^5))$
4. $(root(3)(4))^(-5)$
Which of the following is a reason why, when dealing with an equation or expression, one would switch from using radical notation (square roots, cube roots, etc), to rational exponents? Choose all correct answers.
1. One can use exponent laws to simplify the expression or equation. For example, $root(3)(x^6) * sqrt(x^9)$ becomes $x^(6/3) * x^(9/2) = x^(2+9/2) = x^(4/2 + 9/2) = x^(13/2)$.
2. It avoids the issue of complex numbers. For example, $sqrt(-2)$ is complex, namely $2i$ where $i$ is the square root of -1, but changing this into a rational exponent, $-2^(1/2)$, it is no longer complex.
3. For an expression or equation with division, once the radical has been converted to a rational exponent, the factors can be more easily divided. For example, $sqrt(8x^7) / root(4)(4x^3)$ becomes $(8x^(7/2)) / (4x^(3/4)) = 2x^(7/2 - 3/4) = 2x^(14/4 - 3/4) = 2x^(11/4)$.
4. Using rational exponents can lead to easier forms for evaluation. For example, $root(3)(4^6)$ can be written as $4^(6/3) = 4^2$, which can then be easily evaluated.
Which of the following is a correct way to rewrite the expression $root(10)(3^5) ?$ Choose all that apply.
1. $3^(1/2)$
2. $3^(5/10)$
3. $(3^5)^(1/10)$
4. $(3^10)^(1/5)$
Which of the following is a correct way to rewrite the expression $root(3)(9^(-4)) ?$ Choose all that apply.
1. $(9^(-1/3))^(-4)$
2. $9^(4/3)$
3. $1/(9^(4/3))$
4. $(9^(-4))^(-3)$