Problem 25 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. $$ (-7)^{-3} $$

Step-by-Step Solution

Answer

\((-7)^{-3} = -\frac{1}{343}.

Step 1: Understanding the meaning of a negative exponent

The negative exponent is just the reciprocal of the number with the positive of that exponent, i.e., \(a^{-n} = \frac{1}{a^{n}}.\) Hence, \((-7)^{-3}\) equals \(\frac{1}{(-7)^3}.\)

Step 2: Raising -7 to the power of 3

In the next step, raise -7 to the power of 3. In mathematical notation, this means calculating \((-7)^3.\)

Step 3: Performing the Calculation

Hence, when you calculate \((-7)^3\), you get -343 because -7*-7*-7 is -343. So, the reciprocal of -343 is what \((-7)^{-3}\) equals.

Step 4: Final Result

So, the result of \((-7)^{-3}\) is \(-\frac{1}{343}.\)

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to the power of an exponent. The exponent tells you how many times to multiply the base by itself. For example, if you have the base number 2 and an exponent of 3, denoted as \(2^3\), it means you multiply 2 by itself three times: \(2 \times 2 \times 2 = 8\).

When the exponent is a negative number, it indicates the reciprocal of the base raised to the corresponding positive exponent. So, instead of multiplying, we will be dividing. For instance, \(2^{-3}\) is the same as \(\frac{1}{2^3}\), which equals \(\frac{1}{8}\). Whenever you encounter a negative exponent, remember that you're looking for a fraction where the numerator is the number 1, and the denominator is the base raised to the absolute value of the exponent.

Reciprocal of a Number

The reciprocal of a number is simply the inverse of that number. It's what you multiply a number by to get the multiplicative identity, which is 1. This means if you have a number \(a\), its reciprocal is \(1/a\) or \(a^{-1}\). For example, the reciprocal of 5 is \(1/5\) or \(5^{-1}\).

If dealing with negative numbers, the same rule applies, but you keep the sign. So, the reciprocal of \(-7\) is \(-1/7\). This concept is particularly important when working with negative exponents, as it helps to simplify expressions, allowing us to turn division into multiplication, which can often make solving problems a less complex process.

Mathematical Notation

Mathematical notation is a system of symbols and signs used to represent numbers, operations, and relationships in mathematics. It's a concise and precise language that mathematicians use to avoid ambiguities and to simplify the process of calculations and proofs. For instance, the caret '^' is commonly used to denote exponentiation, so the expression \(a^n\) tells us that 'a' is the base and 'n' is the exponent.

In the context of negative exponents, notation plays a vital role in understanding how to handle the operation. The expression \((-7)^{-3}\) shows the operation of raising -7 to the negative 3rd power. Mastering mathematical notation is key for students as it enables them to read and solve mathematical expressions correctly and efficiently.