Question

Write each expression without parentheses. Assume all variables are positive. $$ \left(\frac{2}{3}\right)^{4} $$

Short answer

Question: Rewrite the given expression without parentheses: $$ \left(\frac{2}{3}\right)^{4} $$ Answer: __The expression without parentheses is: $\frac{16}{81}$.__

Step-by-step solution

Identify the expression inside the parentheses

The expression inside the parentheses is a fraction, \(\frac{2}{3}\).

Apply exponent property to remove parentheses

We can now use the exponent property to simplify the expression. The exponent property tells us that \((\frac{a}{b})^n = \frac{a^n}{b^n}\). In our case, we have: $$ \left(\frac{2}{3}\right)^{4} = \frac{2^4}{3^4} $$

Calculate the powers in numerator and denominator

Now, we need to calculate the powers in both the numerator and the denominator: $$ \frac{2^4}{3^4} = \frac{16}{81} $$

Final Answer

The expression without parentheses is: $$ \left(\frac{2}{3}\right)^{4} = \frac{16}{81} $$

Key concepts

Fraction exponentiation

Fraction exponentiation is a process where you raise a fraction to a power. This means that both the numerator and denominator of the fraction will be raised to the given exponent.

For instance, when you have an expression like \( \left(\frac{2}{3}\right)^4 \), you are essentially asking yourself to multiply both 2 and 3 by themselves four times.

The exponentiation of a fraction can be completed in a few simple steps:

• Recognize the fraction that needs to be exponentiated.
• Apply the power to both the numerator and the denominator separately.
• Simplify the newly formed fraction by performing any necessary calculations.

Knowing how to exponentiate fractions is a valuable skill, as it occurs frequently in algebra and calculus problems. It often presents a clearer path to a solution in more complex expressions.

Exponent properties

Exponent properties are essential rules that simplify mathematical expressions involving powers. They help us manage the potential complexity of working with large or cumbersome numbers. Understanding these properties can make working with expressions like \( \left(\frac{2}{3}\right)^{4} \) much easier.

One fundamental property is that \( (a^m)(a^n) = a^{m+n} \), which allows the combining of like bases when multiplying. Conversely, when dividing, \( \frac{a^m}{a^n} = a^{m-n} \).

Another crucial property is how to handle expressions with a power raised again to a power: \( (a^m)^n = a^{mn} \). In our main example, a key property used is \( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \). This allows us to apply the exponent to both the numerator and the denominator separately.

• These properties simplify seemingly complex problems and guide the rewriting of certain expressions.
• They are foundational to correctly simplifying and solving many algebraic problems.

Simplifying expressions

Simplifying expressions is a key step in solving mathematical problems, particularly when dealing with exponentiation. When you simplify an expression, you're essentially breaking it down into a more understandable and manageable form.

For the example \( \left(\frac{2}{3}\right)^4 \), simplification involves applying the exponent to both parts of the fraction, resulting in \( \frac{2^4}{3^4} \) which simplifies to \( \frac{16}{81} \).

Steps to simplify expressions include:

• Applying relevant exponent rules or properties to make rewriting easier.
• Ensuring that all numerical calculations, such as \( 2^4 \) which equals 16, are completed correctly for both the numerator and the denominator.
• Double-checking the final result for accuracy.

Proper simplification makes further mathematical processes easier, whether you're working on additional operations or attempting to solve equations.