Problem 74 - Elementary and Intermediate Algebra

Question

Use the power of a product rule for exponents to simplify each expression. $$ \left(-\frac{1}{4} t^{3} u^{8}\right)^{2} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{1}{16} \, t^{6} \, u^{16}\).

Step 1: Apply the Power of a Product Rule

The power of a product rule states that \[(a \, b)^{m} = a^{m} \, b^{m}\]Let's apply this to each factor inside the parentheses, \[\left(-\frac{1}{4} t^{3} u^{8}\right)^{2} = \left(-\frac{1}{4}\right)^{2} \, (t^{3})^{2} \, (u^{8})^{2}\]

Step 2: Simplify the Coefficient

For the coefficient part, we need to find \[\left(-\frac{1}{4}\right)^{2}\].Squaring both the numerator and the denominator, we have:\[\left(-\frac{1}{4}\right)^{2} = \frac{1}{16}\]

Step 3: Simplify the Power of t

For \[(t^{3})^{2}\],apply the power of a power rule \((a^{m})^{n} = a^{m \cdot n}\):\[(t^{3})^{2} = t^{6}\]

Step 4: Simplify the Power of u

For \[(u^{8})^{2}\],apply the power of a power rule \((a^{m})^{n} = a^{m \cdot n}\):\[(u^{8})^{2} = u^{16}\]

Step 5: Combine the Results

Now combine all the simplified parts together:\[\frac{1}{16} \, t^{6} \, u^{16}\]

Key Concepts

Simplifying Exponents

In mathematics, exponents are a powerful tool that allows us to express repeated multiplication of a number. Simplifying these exponents is the process of rewriting them in a more manageable form, making calculations easier and more efficient. This usually involves using certain properties of exponents to transform complex expressions into simpler ones.

• The Power of a Product Rule is useful when you have an expression like \( (a \cdot b)^m \), which allows you to distribute the exponent across the product. It simplifies into \( a^m \cdot b^m \).
• We often need to manage fractional coefficients when simplifying, especially if they are raised to a power, e.g., \( \left(-\frac{1}{4}\right)^2 = \frac{1}{16} \). Simplify by applying the power separately to the numerator and the denominator.
• Remember to apply the same power to variables like \( t^3 \) and \( u^8 \) as shown in the example, making sure not to forget multiplying the exponents to simplify them.

Fully understanding these simplification steps allows you to reformat expressions to their simplest forms, helping to avoid common errors and increasing accuracy in problem-solving scenarios.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operators (such as addition and multiplication). These expressions are like math sentences that you can manipulate and simplify.

• When dealing with expressions inside parentheses, address each part separately before putting them back together. This ensures accuracy in complex expressions.
• The presence of variables like \( t \) and \( u \) means the expression is dynamic, changing values based on different inputs.
• Understand that coefficients (e.g., \( -\frac{1}{4} \)) will multiply or divide variables, affecting the entire expression.

Algebraic expressions form the foundation of algebra, enabling us to establish relationships and solve equations by understanding the structure and manipulations possible within them.

Power of a Power Rule

The Power of a Power Rule is a fundamental exponent rule important for simplifying expressions where a term in the base is already an exponent. The rule is articulated as \( (a^m)^n = a^{m \cdot n} \), emphasizing the multiplication of the exponents.

• It's essential to apply this rule in steps to avoid mistakes, especially when multiple variables and terms with exponents are involved.
• For instance, in the expression \( (t^3)^2 \), apply the rule to get \( t^{3 \cdot 2} = t^6 \).
• Once each term is simplified by using this rule, combining all parts will give a clear, simplified result.

By using the Power of a Power Rule, you can transform seemingly complex expressions into their simplest forms, making further algebraic operations or solving equations much more straightforward.