Back to Tutor Intake Assessment: The Real Number System

HSN.RN Tutor Intake — Rational Exponents and Real Number Properties

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Grade 9·12 problems·~13 min·Common Core Math - HS Number and Quantity·domain·rn

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Concepts

We want the power rule $(a^m)^n = a^{mn}$ to hold for ALL
exponents. Based on this requirement, what must $9^{1/2}$ equal?

$9/2 = 4.5$

$\sqrt{9} = 3$

$9^2 = 81$

$2\sqrt{9} = 6$

Which expression is equivalent to $a^{m/n}$ ?

$\sqrt[m]{a^n}$

$\sqrt[n]{a^m}$

$a^m \div a^n$

$n \cdot a^m$

Let $r = p/q$ and $s = m/n$ be any two rational numbers
( $p, q, m, n$ integers, $q \neq 0$ , $n \neq 0$ ). Which statement
correctly explains why $r + s$ is rational?

$r + s = (pn + mq)/(qn)$ . Since $pn + mq$ is an integer and
$qn$ is a nonzero integer, the sum is rational.

$r + s$ must be rational because both terms are positive.

$r + s$ equals a whole number whenever $p = m$ and $q = n$ .

We can verify with examples: $1/2 + 1/3 = 5/6$ , which is
rational, so the sum of any two rationals is rational.

Which of the following expressions is irrational?

$3 + \sqrt{25}$

$\sqrt{4} + \sqrt{9}$

$\frac{1}{2} + \sqrt{7}$

$0.6\overline{6} + \frac{1}{3}$

Compute $0 \times \sqrt{11}$ . Enter a whole number.

Procedures

Evaluate $8^{2/3}$ . Enter a whole number.

Which exponential expression is equivalent to $\sqrt[5]{x^3}$ ?

$x^{5/3}$

$x^{3/5}$

$x^{15}$

$x^{2}$

Which radical expression is equivalent to $y^{4/3}$ ?

$\sqrt[4]{y^3}$

$\sqrt[3]{y^4}$

$4\sqrt[3]{y}$

$\sqrt[3]{4y}$

Simplify $x^{1/2} \cdot x^{1/3}$ . Enter the exponent of $x$ in
the simplified form $x^{?}$ as a fraction in lowest terms
(e.g., enter $5/6$ ).

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