Simplify the following expressions: (2 × 2 × 2)^2/3, (23)^2/3, 22/4d, sqrt(256 × 289), and sqrt((1)/(4…

Question

Simplify the following expressions: (2 × 2 × 2)^2/3, (23)^2/3, 22/4d, sqrt(256 × 289), and sqrt((1)/(4 × 17)).

ScanSolve AI Tutor · Accepted Answer

Here are the solutions to the problems that are clearly visible in the image.

:

(2 \times 2 \times 2)^{\frac{2}{3}}

Simplify the base of the exponent.
$(2 \times 2 \times 2)^{\frac{2}{3}} = (2^3)^{\frac{2}{3}}$
Apply the exponent rule $(a^m)^n = a^{mn}$ .
$(2^3)^{\frac{2}{3}} = 2^{3 \times \frac{2}{3}} = 2^2$
Calculate the final value.
$2^2 = 4$
The answer is $\boxed{4}$ .

:
$(2^3)^{\frac{2}{3}}$
Apply the exponent rule $(a^m)^n = a^{mn}$ .
$(2^3)^{\frac{2}{3}} = 2^{3 \times \frac{2}{3}} = 2^2$
Calculate the final value.
$2^2 = 4$
The answer is $\boxed{4}$ .

:
$\frac{2^2}{4d}$
Calculate the value of the numerator.
$2^2 = 4$
Substitute the value into the expression.
$\frac{2^2}{4d} = \frac{4}{4d}$
Simplify the fraction by canceling the common factor of 4.
$\frac{4}{4d} = \frac{1}{d}$
The answer is $\boxed{\frac{1}{d}}$ .

:
$\sqrt{256 \times 289}$
Use the property $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$ .
$\sqrt{256 \times 289} = \sqrt{256} \times \sqrt{289}$
Calculate the square roots of the individual numbers.
$\sqrt{256} = 16$ $\sqrt{289} = 17$
Multiply the results.
$16 \times 17 = 272$
The answer is $\boxed{272}$ .

:
$\sqrt{\frac{1}{4 \times 17}}$
Use the property $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ .
$\sqrt{\frac{1}{4 \times 17}} = \frac{\sqrt{1}}{\sqrt{4 \times 17}}$
Simplify the numerator and the denominator using $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$ .
$\sqrt{1} = 1$ $\sqrt{4 \times 17} = \sqrt{4} \times \sqrt{17} = 2 \sqrt{17}$
Combine the simplified parts.
$\frac{1}{2 \sqrt{17}}$
Rationalize the denominator by multiplying the numerator and denominator by $\sqrt{17}$ .
$\frac{1}{2 \sqrt{17}} \times \frac{\sqrt{17}}{\sqrt{17}} = \frac{\sqrt{17}}{2 \times 17}$
Simplify the denominator.
$\frac{\sqrt{17}}{34}$
The answer is $\boxed{\frac{\sqrt{17}}{

Simplify the following expressions: (2 x 2 x 2)^(2)/(3), (23)^(2)/(3), (22)/(4d), sqrt(256 x 289), and sqrt((1)/(4 x 17)).

Step-by-step solution

Simplify the base of the exponent.

Apply the exponent rule $(a^m)^n = a^{mn}$ .

Calculate the final value.

Apply the exponent rule $(a^m)^n = a^{mn}$ .

Calculate the final value.

Calculate the value of the numerator.

Substitute the value into the expression.

Simplify the fraction by canceling the common factor of 4.

Use the property $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$ .

Calculate the square roots of the individual numbers.

Multiply the results.

Use the property $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ .

Simplify the numerator and the denominator using $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$ .

Combine the simplified parts.

Rationalize the denominator by multiplying the numerator and denominator by $\sqrt{17}$ .

Simplify the denominator.

Still stuck on this one?

More Mathematics Questions

Simplify the following expressions: (2 x 2 x 2)^(2)/(3), (23)^(2)/(3), (22)/(4d), sqrt(256 x 289), and sqrt((1)/(4 x 17)).

Step-by-step solution

Simplify the base of the exponent.

Apply the exponent rule (am)n=amn(a^m)^n = a^{mn}(am)n=amn.

Calculate the final value.

Apply the exponent rule (am)n=amn(a^m)^n = a^{mn}(am)n=amn.

Calculate the final value.

Calculate the value of the numerator.

Substitute the value into the expression.

Simplify the fraction by canceling the common factor of 4.

Use the property ab=a×b\sqrt{ab} = \sqrt{a} \times \sqrt{b}ab​=a​×b​.

Calculate the square roots of the individual numbers.

Multiply the results.

Use the property ab=ab\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}ba​​=b​a​​.

Simplify the numerator and the denominator using ab=a×b\sqrt{ab} = \sqrt{a} \times \sqrt{b}ab​=a​×b​.

Combine the simplified parts.

Rationalize the denominator by multiplying the numerator and denominator by 17\sqrt{17}17​.

Simplify the denominator.

Still stuck on this one?

More Mathematics Questions

Apply the exponent rule $(a^m)^n = a^{mn}$ .

Apply the exponent rule $(a^m)^n = a^{mn}$ .

Use the property $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$ .

Use the property $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ .

Simplify the numerator and the denominator using $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$ .

Rationalize the denominator by multiplying the numerator and denominator by $\sqrt{17}$ .