Simplifying Cube Roots and Radicals with Variables

• 2

• 4

• 8

• 1

Answer explanation

The cube root of 8, written as \(\sqrt[3]{8}\), is the number that, when multiplied by itself three times, equals 8. Since \(2 \times 2 \times 2 = 8\), the simplified form is 2.

• 6

• 9

Answer explanation

To simplify \( \sqrt[3]{54} \), we factor 54 as \( 27 \times 2 \). Since \( 27 = 3^3 \), we have \( \sqrt[3]{54} = \sqrt[3]{27 \times 2} = \sqrt[3]{27} \cdot \sqrt[3]{2} = 3\sqrt[3]{2} \). Thus, the correct answer is \( 3\sqrt[3]{2} \).

• 2

• 3

• 8

• 16

Answer explanation

To simplify \(\sqrt[3]{32}\), we can express 32 as \(2^5\). Thus, \(\sqrt[3]{32} = \sqrt[3]{2^5} = 2^{5/3} = 2^{1 + 2/3} = 2 \cdot \sqrt[3]{4} = 4\). Therefore, the answer is 4.

• x

Answer explanation

The expression \(\sqrt{x^2}\) simplifies to \(x\) because the square root and the square cancel each other out. However, note that this is true for non-negative values of \(x\). Thus, the correct answer is \(x\).

• y

Answer explanation

The expression \(\sqrt[3]{y^3}\) simplifies to \(y\) because the cube root of \(y^3\) is \(y\). Thus, the correct answer is \(y\).

Answer explanation

To simplify \(\sqrt[3]{z^6}\), we use the property \(\sqrt[3]{a^b} = a^{b/3}\). Thus, \(\sqrt[3]{z^6} = z^{6/3} = z^2\). Therefore, the correct answer is \(z^2\).

• a

Answer explanation

To simplify \(\sqrt[3]{a^9}\), use the property \(\sqrt[3]{x^n} = x^{n/3}\). Here, \(n=9\), so \(9/3=3\). Thus, \(\sqrt[3]{a^9} = a^3\), making \(a^3\) the correct answer.

• b

Answer explanation

To simplify \(\sqrt[3]{b^{12}}\), use the property \(\sqrt[n]{a^m} = a^{m/n}\). Here, it becomes \(b^{12/3} = b^4\). Thus, the correct answer is \(b^4\).

Answer explanation

To simplify \( \sqrt[3]{c^{15}} \), use the property \( \sqrt[n]{a^m} = a^{m/n} \). Here, \( m = 15 \) and \( n = 3 \), so \( \sqrt[3]{c^{15}} = c^{15/3} = c^5 \). Thus, the answer is \( c^5 \).

Answer explanation

To simplify \(\sqrt[3]{d^{18}}\), use the property \(\sqrt[3]{a^b} = a^{b/3}\). Thus, \(d^{18/3} = d^6\). Therefore, the simplified form is \(d^6\), which is the correct answer.