Chapter 8: Problem 27
Write your answer as a power or as a product of powers. $$ (3 \cdot 7)^{2} $$
Short Answer
Expert verified
The expression \((3 \cdot 7)^{2}\) can be expressed as \(9 \times 49 = 441\)
Step by step solution
01
Identify the Base and Exponent
The given expression is \((3 \cdot 7)^{2}\). Here, \(3\) and \(7\) are the bases and \(2\) is the exponent.
02
Apply the Power of a Product Rule
According to the power of a product rule, we can distribute the exponent to both of the bases. So we have \((a \cdot b)^n = a^n \cdot b^n\). Therefore, \((3 \cdot 7)^{2} = 3^2 \cdot 7^2\).
03
Calculate the Powers
Now we calculate each power: \(3^2 = 9\) and \(7^2 = 49\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Exponents
Exponents are the mathematical shorthand for expressing repeated multiplication of the same number. They consist of two parts: the base and the exponent. For example, in the expression \( 3^2 \), 3 is the base and 2 is the exponent. It tells us to multiply 3 by itself a number of times equal to the exponent, so \( 3^2 \) equals \( 3 \times 3 \) or 9. Understanding exponents is fundamental in algebra since they are used to compactly write algebraic expressions and to perform calculations involving powers.
Correctly applying the rules of exponents is crucial. These rules, such as product rule, quotient rule, and power rule, allow us to simplify and manipulate expressions involving powers. For instance, when two numbers with the same base are multiplied, their exponents are added; whereas when a base with an exponent is raised to another exponent, the exponents are multiplied.
Correctly applying the rules of exponents is crucial. These rules, such as product rule, quotient rule, and power rule, allow us to simplify and manipulate expressions involving powers. For instance, when two numbers with the same base are multiplied, their exponents are added; whereas when a base with an exponent is raised to another exponent, the exponents are multiplied.
Base
In an exponential expression, the base is the number that is being repeated in a multiplication task. Referring back to \(3^2\), 3 is once again the base. The base is the foundation of the power, determining the value that is to be multiplied. It is important to identify the base accurately, as it is involved in many exponent rules. For example, the base is crucial in operations like exponentiation and taking roots, and it should remain unchanged when applying such exponent rules.
One common mistake is changing the base when performing exponent operations, which leads to incorrect results. Always keep in mind that the base should stay the same when it is raised to a power or when multiple instances of the base are multiplied or divided.
One common mistake is changing the base when performing exponent operations, which leads to incorrect results. Always keep in mind that the base should stay the same when it is raised to a power or when multiple instances of the base are multiplied or divided.
Power
The power of a number refers to the result of exponentiation, which involves raising a base to an exponent. The power tells us how many times to multiply the base by itself. The term 'power' also refers to the entire expression, such as \( 3^2 \). It is symbolic of the 'strength' or magnitude the base achieves when multiplied by itself a certain number of times.
The application of powers is not limited to just numbers but can extend to variables and other algebraic expressions as well. The concept of powers becomes even more critical as students progress in mathematics, being a key component in formulas, functions, and equations across various topics.
The application of powers is not limited to just numbers but can extend to variables and other algebraic expressions as well. The concept of powers becomes even more critical as students progress in mathematics, being a key component in formulas, functions, and equations across various topics.
Algebraic Expressions
Algebraic expressions are a combination of numbers, variables, and operation signs that stand for a particular quantity. When dealing with powers in algebraic expressions, it's important to identify like terms, which are terms that have the same variable raised to the same power. The expressions can then be simplified by using exponent rules.
In the given exercise, \( (3 \cdot 7)^2 \) is an example of an algebraic expression. Simplifying such expressions using the power of a product rule ensures that the mathematical statement is easier to comprehend and use. Remember, algebraic expressions can represent real-world scenarios, so learning to work with them is not only a mathematical skill but also a practical one.
In the given exercise, \( (3 \cdot 7)^2 \) is an example of an algebraic expression. Simplifying such expressions using the power of a product rule ensures that the mathematical statement is easier to comprehend and use. Remember, algebraic expressions can represent real-world scenarios, so learning to work with them is not only a mathematical skill but also a practical one.