Question

Write your answer as a power or as a product of powers. $$ \left(6 a^{4}\right)^{2}\left(\frac{1}{4} a^{3}\right)^{2} $$

Short answer

The simplified answer as a product of powers is \( \frac{9}{4}a^{14} \).

Step-by-step solution

- Simplify the first part

Start with the first part of the expression, which is \( \left(6 a^{4}\right)^{2} \). Use the power of a power rule, which states that \((a^m)^n = a^{mn}\). Also remember that when a number and a power is raised to another power, each part gets that power, so applying these rules we get: \( \left(6^2\right) \cdot \left(a^{4 \cdot 2}\right) = 36a^{8} \).

- Simplify the second part

Next, simplify the second part \( \left(\frac{1}{4} a^{3}\right)^{2} \). Though it involves a fraction, the same Power of Power rule applies, yielding \( \left(\frac{1}{4}\right)^2 \cdot \left(a^{3 \cdot 2}\right) = \frac{1}{16}a^{6} \).

- Multiply the two parts

Finally, multiply the two parts obtained from above steps: \(36a^{8} \cdot \frac{1}{16}a^{6}\). According to the product of powers rule, you can add the exponents if the bases are the same, in this case, \(a\). Hence, \(36a^{8} \cdot \frac{1}{16}a^{6}\) simplifies to \( \frac{36}{16}a^{8+6} = \frac{9}{4}a^{14} \).

Key concepts

Exponents

Exponents are shorthand for repeated multiplication of the same factor. For instance, the expression \(a^3\) signifies that the base \(a\) is multiplied by itself three times: \(a \times a \times a\). Understanding exponents is fundamental when dealing with algebraic expressions because they help us write and manipulate large numbers or terms in a much more efficient way.

Exponents have several properties and rules that make calculations easier. For example, any number raised to the zero power is equal to 1 (except for the undefined 0^0), and a negative exponent signifies division by that base raised to the positively opposite exponent. It's essential to grasp these properties, as they are the building blocks for more complex concepts like the power of a power rule.

Simplifying Algebraic Expressions

Simplifying algebraic expressions can make them easier to understand and work with. This process involves combining like terms, factoring, expanding expressions, and applying various algebraic rules. When we simplify an expression, we aim to write it in the most concise and efficient way possible, often with the least number of terms.

Applying Exponent Rules

The understanding of exponent rules is crucial in simplifying expressions involving power. When simplifying, always remember to look for like bases and apply relevant exponent rules to condense the expression. Simplified expressions are less prone to errors and are easier to evaluate or substitute values into, which is often required in algebraic problems.

Product of Powers Rule

The product of powers rule is an essential concept in algebra that makes it easier to multiply two exponential terms with the same base. According to the rule, you simply add the exponents while keeping the base the same: \(a^m \times a^n = a^{m+n}\). This rule comes in extremely handy when dealing with polynomials or simplifying large expressions.

Applying the Rule

When faced with a product of powers during simplification, always look to combine the exponents of like bases. This not only shortens the expressions but also paves the way for further simplification. Students should practice this rule to gain confidence in manipulating and simplifying algebraic expressions, as it is a commonly encountered situation in algebra.