What is the value of 16^(3/4)?

In a rational exponent the denominator gives the root and the numerator gives the power. So 16^(3/4) is the fourth root of 16, raised to the third power. The fourth root of 16 is 2 because 2 × 2 × 2 × 2 = 16, and 2³ = 8. Taking the power first gives the same answer by a harder route: 16³ = 4,096, and the fourth root of 4,096 is 8. The distractor 12 comes from multiplying 16 by 3/4, and 64 comes from cubing 4 instead of 2.

Which expression is equivalent to (x^(2/3)) · (x^(1/2))?

x^(7/6). The product rule for exponents applies to fractional exponents exactly as it does to whole-number exponents: multiplying powers of the same base adds the exponents. Adding 2/3 and 1/2 requires the common denominator 6: 4/6 + 3/6 = 7/6, so the product is x^(7/6). The distractor x^(1/6) comes from subtracting the exponents, which is the quotient rule, and x^(2/6) comes from multiplying the exponents, which is the power-of-a-power rule.

Between which two consecutive tenths does √62 lie?

Between 7.8 and 7.9. First trap the value between whole numbers: 7² = 49 and 8² = 64, so √62 lies between 7 and 8, and much closer to 8 since 62 is near 64. Now narrow to tenths: 7.8² = 60.84, which is less than 62, and 7.9² = 62.41, which is greater than 62. Therefore 7.8 < √62 < 7.9. The choice 6.2 to 6.3 confuses the radicand with the root itself, a frequent error on estimation items.

Radicals, Rational Exponents, & Approximatin | Free Guide 2026

Radicals, Rational Exponents, & Approximating Irrational Numbers

Section 5.3 introduced square roots, cube roots, and the laws of exponents separately. This section joins them, which is exactly what the Level A standard requires: "rewrite expressions involving radicals and rational exponents using the properties of exponents."

Radicals Are Exponents

A rational exponent — an exponent that is a fraction — is another way of writing a root.

x^{1/2} = \sqrt{x} \qquad x^{1/3} = \sqrt[3]{x} \qquad x^{1/n} = \sqrt[n]{x}

When the numerator is not 1, it becomes a power applied to the root:

x^{m/n} = \sqrt[n]{x^m} = \left(\sqrt[n]{x}\right)^m

The denominator is the root; the numerator is the power. A mnemonic that survives test pressure: the root is down below.

Rational exponent form	Radical form	Value
$16^{1/2}$	$\sqrt{16}$	4
$27^{1/3}$	$\sqrt[3]{27}$	3
$8^{2/3}$	$\left(\sqrt[3]{8}\right)^2 = 2^2$	4
$81^{3/4}$	$\left(\sqrt[4]{81}\right)^3 = 3^3$	27
$25^{-1/2}$	$\dfrac{1}{\sqrt{25}}$	$\tfrac{1}{5}$

Always take the root first. $8^{2/3}$ computed as $\sqrt[3]{64}$ also gives 4, but on non-calculator items rooting first keeps the numbers small: $\sqrt[3]{8} = 2$ , then $2^2 = 4$ .

The Exponent Laws Still Apply

Nothing about fractional exponents changes the rules. That is the whole reason the notation exists.

Law	Rule	Fractional example
Product	$x^a \cdot x^b = x^{a+b}$	$x^{1/2} \cdot x^{1/3} = x^{5/6}$
Quotient	$x^a \div x^b = x^{a-b}$	$x^{3/4} \div x^{1/4} = x^{1/2} = \sqrt{x}$
Power of a power	$(x^a)^b = x^{ab}$	$\left(x^{2/3}\right)^{3} = x^{2}$
Negative	$x^{-a} = \dfrac{1}{x^a}$	$x^{-1/2} = \dfrac{1}{\sqrt{x}}$
Zero	$x^0 = 1$	$\left(\sqrt{7}\right)^0 = 1$

Worked example. Simplify $\dfrac{\sqrt{x} \cdot \sqrt[4]{x}}{x^{1/4}}$ .

= \frac{x^{1/2} \cdot x^{1/4}}{x^{1/4}} = x^{1/2 + 1/4 - 1/4} = x^{1/2} = \mathbf{\sqrt{x}}

Simplifying Radicals

A radical is in simplest form when no perfect-square factor remains under the sign. Pull out the largest perfect square you can find.

\sqrt{72} = \sqrt{36 \cdot 2} = \sqrt{36} \cdot \sqrt{2} = \mathbf{6\sqrt{2}}

\sqrt{200} = \sqrt{100 \cdot 2} = \mathbf{10\sqrt{2}}

\sqrt[3]{54} = \sqrt[3]{27 \cdot 2} = \mathbf{3\sqrt[3]{2}}

Radicals with the same radicand combine like terms: $6\sqrt{2} + 10\sqrt{2} = 16\sqrt{2}$ . Radicals with different radicands do not: $\sqrt{2} + \sqrt{3}$ cannot be simplified further, and it is emphatically not $\sqrt{5}$ .

Rational vs. Irrational Numbers

	Rational	Irrational
Definition	can be written $\frac{a}{b}$ with integers $a, b \ne 0$	cannot be written as such a ratio
Decimal	terminates or repeats	never terminates, never repeats
Examples	$\tfrac{3}{8} = 0.375$ ; $0.\overline{6}$ ; $-4$ ; $\sqrt{49}$	$\sqrt{2}$ , $\sqrt{10}$ , $\pi$ , $\sqrt[3]{5}$

Two traps TABE exploits:

$\sqrt{\text{perfect square}}$ is rational. $\sqrt{49} = 7$ is an integer, hence rational. Only roots of non-perfect powers are irrational.
$\frac{22}{7}$ is rational. It is a ratio, so it is rational by definition, even though it approximates the irrational $\pi$ .

Locating an Irrational Number

The tested method is trapping between perfect squares, then narrowing — exactly the procedure the DRC specification describes for truncating a decimal expansion.

Where does $\sqrt{2}$ live?

$1^2 = 1$ and $2^2 = 4$ , so $1 < \sqrt{2} < 2$ .
Try tenths: $1.4^2 = 1.96$ and $1.5^2 = 2.25$ , so $1.4 < \sqrt{2} < 1.5$ .
Try hundredths: $1.41^2 = 1.9881$ and $1.42^2 = 2.0164$ , so $1.41 < \sqrt{2} < 1.42$ .

Each round adds one decimal place. Two rounds is plenty for any TABE item.

Where does $\sqrt{55}$ sit? $7^2 = 49$ and $8^2 = 64$ , so it is between 7 and 8, and because 55 is nearer 49 than 64, it is a little below the midpoint: $\sqrt{55} \approx 7.4$ .

Ordering a mixed list

Order $\sqrt{10}$ , $3.2$ , $\pi$ , and $\tfrac{16}{5}$ from least to greatest.

Value	Decimal
$\pi$	3.14159…
$3.2$	3.2
$\tfrac{16}{5}$	3.2
$\sqrt{10}$	3.162…

So: $\pi < \sqrt{10} < 3.2 = \tfrac{16}{5}$ . Converting everything to a decimal with three places is the reliable way to order a mixed list under time pressure.

Estimating expressions containing irrationals

The specification names $\pi^2$ as an example. Replace the irrational with its bounds and square:

3.1 < \pi < 3.2 \;\Rightarrow\; 9.61 < \pi^2 < 10.24

So $\pi^2 \approx 9.9$ — close enough to identify the right answer choice without a calculator.

TABE Mathematics

5.6 Radicals, Rational Exponents, & Approximating Irrational Numbers

Key Takeaways