Exponents and Roots MCQs with Answers & Explanations

The cube root operation finds a number that multiplies by itself three times to get the original number. Since 4 * 4 * 4 = 64, the cube root is 4.

We need to find a number whose cube is 216. Since 6 * 6 * 6 equals exactly 216, the cube root of 216 is 6.

The cube root of 343 is a number that, when raised to the third power, yields 343. That number is 7, because 7^3 = 343.

To evaluate the cube root of 512, we identify the number that gives 512 when cubed. Since 8 * 8 * 8 = 512, the answer is 8.

The cube root of 1000 requires finding the base that creates 1000 when multiplied by itself three times. 10 * 10 * 10 equals 1000, so the answer is 10.

To simplify sqrt(50), find the largest perfect square factor of 50, which is 25. Rewrite sqrt(50) as sqrt(25 * 2), which simplifies to 5*sqrt(2).

The largest perfect square that divides 27 is 9. We can write sqrt(27) as sqrt(9 * 3). The square root of 9 is 3, making the simplified form 3*sqrt(3).

Find the highest perfect square factor of 72, which is 36. Since 72 = 36 * 2, we take the square root of 36 to get 6 outside, leaving 6*sqrt(2).

The number 48 can be factored into 16 * 3. Because 16 is a perfect square, its square root (4) can be brought outside the radical, resulting in 4*sqrt(3).

Factor 20 into a perfect square and another number, which is 4 * 5. Taking the square root of 4 gives 2, which leaves us with the simplified form 2*sqrt(5).

Look for a perfect cube factor of 16, which is 8. Writing cbrt(16) as cbrt(8 * 2) allows us to bring out the cube root of 8 (which is 2), yielding 2*cbrt(2).

The number 54 can be divided by the perfect cube 27 (27 * 2 = 54). Evaluating the cube root of 27 gives 3, so the expression simplifies to 3*cbrt(2).

Separate x^3 into x^2 * x. Since x^2 is a perfect square, taking its square root gives x on the outside, leaving one x inside the radical: x*sqrt(x).

Rewrite y^5 as the product of the largest perfect square (y^4) and y. The square root of y^4 is y^2, bringing it outside the radical to get y^2*sqrt(y).

Evaluate the numerical and variable parts separately. The square root of 16 is 4, and the square root of x^4 is x^2. This gives the exact simplified form 4x^2.

You can add radicals just like like terms when the numbers inside the roots are identical. Adding an implicit 1*sqrt(2) to 3*sqrt(2) yields 4*sqrt(2).

Because both terms have sqrt(3), they are like radicals. Simply subtract the coefficients: 5 - 2 = 3, maintaining the radical to get 3*sqrt(3).

Since the radicands (number under the root) are the same, treat sqrt(5) as a variable. Add the coefficients 2 and 4 to get 6, giving a result of 6*sqrt(5).

The second term has an implicit coefficient of 1. Subtracting 1 from 7 gives 6. The radical part remains the same, producing 6*sqrt(7).

First simplify each radical: sqrt(12) is 2*sqrt(3) and sqrt(27) is 3*sqrt(3). Now that they have the same radicand, add them: 2*sqrt(3) + 3*sqrt(3) = 5*sqrt(3).