Problem 6 - Algebra 1

Question

Match the power with the words that describe it. A. five to the sixth power B. two to the fifth power C. five squared D. five cubed $$ 2^{5} $$

Step-by-Step Solution

Answer

Here are the matches: \n A matches with \(5^{6}\), \n B matches with \(2^{5}\), \n C matches with \(5^{2}\), and \n D matches with \(5^{3}\).

Step 1: Understand the Word Descriptions for Powers

First, understand what the word descriptions mean. 'To the nth power' means a number is multiplied by itself n times. 'Squared' means a number is multiplied by itself once, and 'cubed' means a number is multiplied by itself two times.

Step 2: Match the Descriptions

After understanding what each word description means, match them to given power expressions. \n A. 'five to the sixth power' matches with \(5^{6}\), \n B. 'two to the fifth power' matches with \(2^{5}\), \n C. 'five squared' matches with \(5^{2}\), and \n D. 'five cubed' matches with \(5^{3}\).

Step 3: Verification

Verify your answers by checking if the description and the given power expressions mathematically fit. \n For example, 'two to the fifth power' does match with the power \(2^{5}\).

Key Concepts

Powers of Numbers

A power or exponent indicates how many times a number, known as the base, is multiplied by itself. In mathematical notation, this is represented as \(a^n\), where \(a\) is the base and \(n\) is the exponent or power. This notation helps simplify the process of repeated multiplication, making complex calculations easier to manage.

• For example, \(5^6\) means 5 is multiplied by itself six times: \(5 \times 5 \times 5 \times 5 \times 5 \times 5\).
• The expression \(2^5\) requires multiplying 2 by itself five times: \(2 \times 2 \times 2 \times 2 \times 2\).

Understating powers also allows us to communicate these ideas efficiently in both written and spoken mathematics, avoiding lengthy expressions. When identifying powers, always consider the number of times the base is used as a factor.

Square and Cube

The terms 'square' and 'cube' are special names in mathematics for specific powers. They indicate multiplying a number by itself a certain number of times. Squaring a number means raising it to the second power, or multiplying it by itself once. Similarly, cubing a number refers to raising it to the third power, multiplying it by itself twice.

• "Five squared" is simply \(5^2\), or \(5 \times 5 = 25\).
• "Five cubed" refers to \(5^3\), or \(5 \times 5 \times 5 = 125\).

Recognizing these terms is crucial, as they frequently appear in both geometry and algebra. Squaring relates to the area of squares, whereas cubing is relevant in finding the volume of cubes. Hence, understanding these concepts is building a foundation for more complex mathematical operations. This knowledge assists in efficiently conveying and solving mathematical problems.

Multiplication of Numbers

Multiplication is a fundamental arithmetic operation representing repeated addition of the same number. This concept extends naturally when dealing with powers. Instead of adding a number over and over, we multiply it based on its power. For example, understanding multiplication is crucial for calculating expressions like \(2^5\).

• This becomes \(2 \times 2 = 4\), \(4 \times 2 = 8\), \(8 \times 2 = 16\), and finally, \(16 \times 2 = 32\).

The mastery of multiplication supports efficiency when handling powers, ensuring accurate computations. It serves as a means to consolidate concepts such as exponents, squares, and cubes in a cohesive manner, not just theoretically but practically, too. Moreover, the multiplication rules apply irrespective of the types of numbers, whether they're whole numbers, decimals, or fractions, allowing for broad applications in various fields.