Problem 42 - Algebra 1

Question

Rewrite the expression with positive exponents. $$\left(6 a^{-3}\right)^{3}$$

Step-by-Step Solution

Answer

The expression \( (6 a^{-3})^3 \) can be rewritten with positive exponents as \( \frac{216}{a^9} \).

Step 1: Identify Exponent Rules

Understanding exponent rules is key for solving this problem. A power of a power rule says that \( (a^m)^n = a^{m*n} \) and a negative exponent rule mentions that \( a^{-n} = \frac{1}{a^n} \). These rules will be used to simplify the expression.

Step 2: Applying the power of a power rule

By applying the power of a power rule to the expression \( (6 a^{-3})^3 \), we get \(6^3 * a^{-3*3} = 216 * a^{-9}\).

Step 3: Applying the negative exponent rule

By applying the negative exponent rule, \( a^{-9} = \frac{1}{a^9}\). Replace \( a^{-9} \) with \( \frac{1}{a^9} \) in the expression obtained from Step 2. We get: \( 216 * \frac{1}{a^9}\).

Key Concepts

Exponent Rules

Exponent rules are fundamental principles that help us manipulate and simplify expressions involving powers. There are several rules, but some are more commonly used in basic algebra.

• Product of Powers Rule: This rule states that when you multiply two powers with the same base, you can add the exponents. The formula is given by: \( a^m \cdot a^n = a^{m+n} \).
• Quotient of Powers Rule: When dividing two powers with the same base, subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
• Power of a Power Rule: This rule helps when raising a power to another power. It states: \( (a^m)^n = a^{m*n} \).
• Negative Exponent Rule: If you have a negative exponent, move the base to the denominator and make the exponent positive: \( a^{-n} = \frac{1}{a^n} \).
• Zero Exponent Rule: Any non-zero base with an exponent of zero equals one: \( a^0 = 1 \).

By using these rules, we can break down complex expressions into simpler forms, making them easier to solve or transform.

Power of a Power Rule

The power of a power rule is very handy when you encounter an expression where a base with an exponent is raised to another power. The rule states that you can multiply the exponents together.For example, in the expression \( (a^m)^n \), you would apply the power of a power rule and simplify it to \( a^{m \cdot n} \). This rule tells us to take each exponent and combine them into a single one by multiplication.
For the problem given, applying the power of a power rule to \((6 a^{-3})^3\) allows us to deal with each part separately:

• For the constant 6 raised to the power of 3: \(6^3 = 216\).
• For the variable \(a\) raised to the power of -3, and then 3: Use the power of a power rule to multiply these exponents: \((-3) \times 3 = -9\).

This simplification gives us \(216 \cdot a^{-9}\), making the calculation more straightforward.

Negative Exponent Rule

Negative exponents might look intimidating at first, but they are simple to handle once you understand the rule. The negative exponent rule allows us to rewrite negative exponents as positive by taking the reciprocal.When you see a negative exponent, such as \(a^{-n}\), you convert it into a positive by rewriting it as \(\frac{1}{a^n}\). This transformation helps to write expressions with only positive exponents, which are often more convenient for further calculations.In the example \(216 \cdot a^{-9}\) from the previous section:

• The part \(a^{-9}\) becomes \(\frac{1}{a^9}\) when applying the negative exponent rule.
• Thus, the expression simplifies to \(216 \times \frac{1}{a^9}\).

By transforming all negative exponents into positive ones, complicated expressions become simpler to understand and solve, paving the way for easier algebraic manipulations.