Problem 48 - Intermediate Algebra

Question

Simplify each expression. Write answers using positive exponents. $$ \left(\frac{g^{20}}{t^{30}}\right)^{-4} $$

Step-by-Step Solution

Answer

\( \frac{t^{120}}{g^{80}} \)

Step 1: Apply Negative Exponent Rule

When an expression is raised to a negative exponent, take the reciprocal and make the exponent positive. So, for our expression \( \left(\frac{g^{20}}{t^{30}}\right)^{-4} \), we write it as: \( \left(\frac{t^{30}}{g^{20}}\right)^{4} \).

Step 2: Distribute the Exponent

Distribute the exponent 4 to both the numerator and the denominator of the fraction. This means we raise \( t^{30} \) to the 4th power and \( g^{20} \) to the 4th power: \( \frac{t^{30 \times 4}}{g^{20 \times 4}} \).

Step 3: Simplify the Powers

Compute the exponents in the numerator and the denominator. Multiply the exponents: \( t^{30 \times 4} = t^{120} \) and \( g^{20 \times 4} = g^{80} \).

Step 4: Write the Simplified Expression

Combine the results into one expression: \( \frac{t^{120}}{g^{80}} \). This is the expression using positive exponents.

Key Concepts

Negative Exponents

Negative exponents might seem tricky, but they offer a straightforward insight into multiplication and division of powers. When you see a negative exponent, like in the expression \(x^{-n}\), it means you should take the reciprocal, or the inverse, of the base raised to a positive exponent.
For example, \(x^{-3}\) is equivalent to \(\frac{1}{x^3}\).

• This rule helps in transforming expressions into forms that are easier to work with, especially for division and multiplication.


• In our exercise, the negative exponent \(-4\) converts \( \left(\frac{g^{20}}{t^{30}}\right)^{-4} \) to \( \left(\frac{t^{30}}{g^{20}}\right)^{4} \) by flipping the fraction and turning the exponent positive.

Understanding negative exponents allows us to simplify complex expressions and is an essential skill in algebra.

Reciprocal

The reciprocal is simply the inverse of a number or expression. For fractions, the reciprocal is found by swapping the numerator and the denominator. If you have a fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \).
This concept is crucial when dealing with negative exponents as it helps in rewriting expressions with positive exponents.

• In the given exercise, the reciprocal of the expression \( \frac{g^{20}}{t^{30}} \) is \( \frac{t^{30}}{g^{20}} \), achieved by flipping the fraction.


• Before addressing the powers, taking the reciprocal aligns with changing the negative exponent to positive, simplifying calculations.

Grasping the reciprocal easily aids the transformation of expressions and enables successful manipulation in solving problems.

Power of a Power

This rule is used when an exponent itself is raised to another power. In such cases, you simply multiply the exponents. For instance, the expression \((a^m)^n\) results in \(a^{m \times n}\).
This is a fundamental principle in algebra, especially useful in simplifying expressions.

• In solving our exercise, after flipping to get \( \left(\frac{t^{30}}{g^{20}}\right)^{4} \), we apply this rule to distribute the power of 4 across both \(t^{30}\) and \(g^{20}\).


• Calculating \(t^{30\times4}\) gives \(t^{120}\) and \(g^{20\times4}\) gives \(g^{80}\).

Understanding how to correctly use this exponent rule is essential for simplifying expressions efficiently.

Simplifying Expressions

Simplifying expressions involves breaking down complex algebraic terms into their simplest forms by applying mathematical rules. This often includes using laws of exponents, like handling negative exponents and finding reciprocals, to achieve clarity in the expression.
In our exercise, after calculating the powers in the numerator and denominator, the expression is simplified to \(\frac{t^{120}}{g^{80}}\).

• This final simplified form is easier to interpret and use for further algebraic operations such as addition, subtraction, or substitution.


• Simplifying also means ensuring all exponents are positive, making handling fractions and further calculations much more manageable and accurate.

Effective simplification is integral to problem-solving, ensuring that expressions are clear and ready for further mathematical processes.