Question

\((\sqrt[3]{x})^{3}=\) __________.

Short answer

x

Step-by-step solution

Understand the Cube Root Concept

The cube root of a number is a value that, when multiplied by itself three times, gives the original number. Mathematically, the cube root of a number x is written as \(\root{3}\rt{x} \).

Recognize the Power Rule

When raising a base to a power, and then taking the same root of that base, the root and power cancel each other out. Here it means \((\root{3}\rt{x})^{3} \) simplifies directly to x.

Apply the Cancellation

Apply the rule in the given problem: \((\root{3}\rt{x})^{3} \). This simplifies to x, as raising a number to a power and then taking the same root cancels each other out.

Key concepts

Power Rule

The power rule is a basic principle in algebra. It helps simplify expressions where a base number is raised to an exponent. If you have a base number, like x, and raise it to an exponent, like 3, and then take the cube root (\root{3}\rt{}), you end up with the original base. So, \((\root{3}\rt{x})^{3} \) simplifies to x.

The rule can be written as \((a^{m})^{n} = a^{m \times n} \).

Here are a few examples to help you understand better:

• \((x^{2})^{3} = x^{2 \times 3} = x^{6}\),
• \((y^{1/2})^{4} = y^{1/2 \times 4} = y^{2}\),
• \((z^{3})^{1/3} = z^{3 \times 1/3} = z^{1} = z\).

When you use roots instead of exponents, the concept is the same. The power and root cancel each other out.

Simplification of Radicals

Radicals are often tricky to simplify. But knowing a few key rules can make it easier.

A radical, like \(\root{n}\rt{x} \), represents the root of a number. For example, the square root of 9 is 3 because \3 \times 3 = 9\. Similarly, the cube root of 8 is 2 because \2 \times 2 \times 2 = 8\.

Simplifying radicals involves reducing them to their simplest form. Here's how:

• Find the prime factors of the number inside the radical.
• Group the factors into pairs or triplets (depending on square or cube roots).
• Bring one factor out of the radical for each pair or triplet formed.

For example, simplify \(\root{3}\rt{27} \):

• Prime factors of 27: \3 \times 3 \times 3\.
• Group into triplets: \ (3 \times 3 \times 3) \.
• Cube root of 27 is 3.

So, \(\root{3}\rt{27} \) simplifies to 3.

Exponents

Exponents are a way to express repeated multiplication. For example, \x^{3} = x \times x \times x\.

Here are some key points about exponents:

• \( x^{1} = x\)
• \( x^{0} = 1\)
• When multiplying like bases, add the exponents: \ x^{a} \times x^{b} = x^{a+b} \.
• When dividing like bases, subtract the exponents: \ x^{a} \times x^{b} = x^{a-b} \.

Understanding these rules can help simplify complex expressions.

For instance, simplify \ ( x^{3} \times x^{4} ) \:

• Add the exponents: \ 3 + 4 = 7 \.
• The simplified form is \ x^{7} \.

Another example is simplifying \ ( x^{5} / x^{2} ) \:

• Subtract the exponents: \ 5 - 2 = 3 \.
• The simplified form is \ x^{3} \.

With these rules in mind, tackling exponents in any problem becomes easier.