Problem 5 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. $$ \frac{1}{4^{-3}} $$

Step-by-Step Solution

Answer

64

Step 1: Understanding Negative Exponents

A negative exponent means to divide by that number, rather than multiply. More formally, \(a^{-n} = 1/a^n\), where 'a' is the base and '-n' is the negative exponent. In this exercise the base is '4', and the negative exponent is '3'.

Step 2: Apply the Negative Exponent Rule

Applying the rule to \(\frac{1}{4^{-3}}\) we obtain \(\frac{1}{1/4^{3}}\), which simplifies to \(\frac{1}{1/64}\).

Step 3: Final Simplification

The further simplification of \(\frac{1}{1/64}\) leads to '64'. Because dividing by a fraction is equivalent to multiplying by its reciprocal which is '64' in this case.

Key Concepts

Exponent Rules

Negative exponents might seem tricky at first, but they actually follow straightforward principles. The key idea behind a negative exponent is that it signifies the reciprocal of the base raised to the opposite positive exponent. For example, with the expression \( 4^{-3} \), the negative exponent indicates that instead of multiplying 4 by itself, we should consider the reciprocal of \( 4^{3} \).
To break it down:

• Start with the base, which is 4.
• The negative sign tells us to take the reciprocal.
• Then, apply the positive exponent, in this case, 3, to get \( \frac{1}{4^3} \).

So, \( 4^{-3} \) becomes \( \frac{1}{4^{3}} \), which simplifies the evaluation of expressions involving negative exponents.

Fraction Simplification

Simplifying fractions is a vital skill in math, helping to make expressions clearer and more manageable. When dealing with fractions that involve negative exponents, it's helpful to understand how dividing by a fraction works. Recall that dividing by a fraction is the same as multiplying by its reciprocal.
Let's look at the expression \( \frac{1}{4^{-3}} \). We learned how the negative exponent transforms \( 4^{-3} \) into \( \frac{1}{4^3} \). Therefore, this expression becomes \( \frac{1}{\frac{1}{64}} \).
To simplify further, multiply by the reciprocal of \( \frac{1}{64} \), which results in:

• Step 1: Recognize that \( \frac{1}{\frac{1}{64}} = 64 \).
• Step 2: Multiplying by the reciprocal simplifies the fraction as dividing by it.

This makes simplifying fractions with negative exponents much easier, turning them into more straightforward whole numbers or simpler fractions.

Mathematical Expressions

Mathematical expressions can often involve complex operations, but understanding them involves breaking them into smaller, understandable parts. Every mathematical expression is essentially a combination of numbers, variables, and operators (like addition, multiplication, etc.) that collectively represent a value or a relationship.
Consider the expression \( \frac{1}{4^{-3}} \). At first glance, this might look complicated, but once we recognize the influence of the negative exponent and fraction rules, it becomes simpler.

• Break down the expression by examining each component separately.
• Use knowledge of exponents and fractions to simplify the equation step by step.
• Combine the steps to solve the expression entirely.

Mastering the understanding of how to deconstruct and solve such expressions is pivotal for dealing with more advanced mathematical problems in the future, ensuring a solid grasp of fundamental principles.