Problem 33 - Algebra 1: Concepts and Skills

Question

Evaluate the expression without using a calculator. $$ \left(4^{-1}\right)^{-3} $$

Step-by-Step Solution

Answer

The expression \((4^{-1})^{-3}\) evaluates to 64.

Step 1: Understand Negative Exponents

Negative exponents represent reciprocals. So, \(4^{-1}\) is equivalent to \(1/4\). Let's replace \(4^{-1}\) in our expression with \(1/4\). Now, our expression becomes \((1/4)^{-3}\).

Step 2: Apply Power of a Power rule

According to the rule, when you raise a power to a power, you just multiply the exponents. Our base now is \(1/4\) and it is raised to the power of \(-3\). This implies that we have to take the reciprocal of the base three times.

Step 3: Evaluate the Expression

When you take the reciprocal of \(1/4\) once, you get \(4/1\) or \(4\). When you do it three times, you get \(4*4*4 = 64\). So, \((1/4)^{-3} = 64\).

Key Concepts

Negative Exponents

Understanding negative exponents is essential when working with various algebraic expressions. Negative exponents denote the reciprocal of a base raised to a positive exponent. In simpler terms, if you have an expression like \(a^{-n}\), where \(a\) is a non-zero base and \(n\) is a positive integer, the negative exponent tells you to take the reciprocal of the base and then raise it to the corresponding positive exponent. For example, \(4^{-1}\) equates to \(1/4\), because you take the reciprocal of 4 (which is \(1/4\)) and raise it to the first power. Essentially, it's a way of expressing division as an exponent.

It's crucial to note that any number, except zero, raised to a negative exponent will result in a fraction. A common mistake is to think that negative exponents create negative numbers. In fact, they do not affect the sign of the base number; only its position as a numerator or denominator changes. So in practice, to evaluate an expression like \(4^{-1}\), we rewrite it as the reciprocal \(1/4\), and to tackle something like \((4^{-1})^{-3}\), we begin by understanding that it involves taking the reciprocal of \(4^{-1}\), which is 4, and then dealing with the exponent -3 in the next steps.

Power of a Power Rule

The power of a power rule is an exponentiation property that comes in handy when dealing with complex algebraic operations. This rule states that if you have a power raised to another power, like \((a^m)^n\), you can simplify the expression by multiplying the exponents: \(a^{m*n}\). It simplifies the process of exponentiation by turning a potentially cumbersome calculation into a more manageable one. For instance, if you have \((4^{-1})^{-3}\), applying this rule would suggest that you multiply the exponents -1 and -3.

This operation simplifies our expression to \(4^{(-1)*(-3)}\) or \(4^3\). Employing this rule accurately requires a firm grasp of how to deal with negative numbers when multiplying, as two negatives make a positive. As a result, raising \(4^{-1}\) to the power of -3 actually results in a positive exponent which makes our computation much smoother.

Reciprocals in Algebra

A reciprocal in algebra is simply the inverse of a number or expression. When you have a non-zero number \(a\), its reciprocal is \(1/a\). This concept is critical when working with negative exponents, as they often require you to find the reciprocal of a number. Reciprocals are used to transform division problems into multiplication ones, which are typically easier to handle.

For example, the reciprocal of 4 is \(1/4\), and the reciprocal of \(1/4\) is 4. In algebraic expressions, reciprocals are helpful in simplifying fractions, solving equations, and dealing with rates or ratios. They are also central to understanding the idea behind negative exponents, as a negative exponent essentially instructs us to use the reciprocal of the base.

Exponentiation

Exponentiation is the operation of raising a number, known as the base, to a power, which is the exponent. It represents repeated multiplication, where the base is multiplied by itself as many times as indicated by the exponent. For instance, \(4^3\) would signify 4 multiplied by itself three times: \(4 \times 4 \times 4\), resulting in 64. Exponents can be positive, negative, or even fractions, each type lending to different rules and implications on the base number.

Understanding exponentiation allows us to manipulate and simplify expressions with greater ease. For expressions with negative exponents, one needs to recognize that the operation involves reciprocal operations before undergoing the multiplication process. Therefore, being comfortable with exponentiation is necessary to fluently navigate through various algebraic problems and find solutions efficiently.