Problem 37 - Algebra 1: Concepts and Skills

Question

Copy and complete the statement. $$ \left(\frac{a^{2}}{b}\right)^{5}=\frac{a^{?}}{b^{5}} $$

Step-by-Step Solution

Answer

The completed statement is \(\left(\frac{a^{2}}{b}\right)^{5} = \frac{a^{10}}{b^{5}}\).

Step 1: Identify the Exponent Rule

The rule applicable in this situation is \((a^{m})^{n} = a^{mn}\). In plain language, this rule states that when you have a power to a power, you multiply the exponents to find the equivalent single exponent.

Step 2: Apply the Exponent Rule

Applying the exponent rule to the given equation: \(\left(\frac{a^{2}}{b}\right)^{5} = \frac{a^{2*5}}{b^{5}}\). Multiply 2 and 5 to find the power of a.

Step 3: Complete the Equation

By multiplying the exponents, we determine that \(\left(\frac{a^{2}}{b}\right)^{5} = \frac{a^{10}}{b^{5}}\). The question mark in the original equation should be replaced with 10 to complete the equation correctly.

Key Concepts

Exponentiation

Exponentiation is one of the core operations in mathematics, going beyond basic addition, subtraction, multiplication, and division. It involves raising a number, known as the "base," to a power, which is represented by a number called the "exponent." The operation answers the question of how many times we multiply the base by itself. For instance, in the expression \(a^2\), "\(a\)" is the base, and "2" is the exponent, telling us to multiply \(a\) by itself once. Hence, \(a^2 = a \times a\).

Exponentiation is a fundamental concept used in various fields like science, engineering, and finance. It simplifies expressions and equations, enabling quicker computations and transformations. Mastery of exponentiation helps us in understanding more complex mathematical concepts such as roots and logarithms.

Power of a Power Property

The Power of a Power Property is a rule of exponents that dramatically simplifies expressions involving multiple exponents. When you take an exponentiated term and raise it to another exponent, you can multiply the exponents. This property is formally expressed as \((a^m)^n = a^{m \times n}\).

This property becomes especially useful in algebra when simplifying expressions or solving equations that involve powers. For example, consider the expression \((a^2)^5\). By applying the Power of a Power Property, we multiply 2 by 5, yielding \(a^{10}\). This simplification saves computation time and effort, particularly when dealing with larger numerical values for exponents. Understanding and applying this property is essential for efficiently solving algebraic problems.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations (such as addition, subtraction, multiplication, division, and exponentiation). They form the backbone of algebra, allowing us to represent complex mathematical ideas succinctly.

An algebraic expression like \(\left(\frac{a^2}{b}\right)^5\) consists of several components: "\(a^2\)" as a term raised to an outer exponent 5, and a division by "\(b\)." To simplify such expressions, understanding the rules of exponents, like the Power of a Power Property, is crucial. In this expression, applying the exponent rules allows us to simplify it to \(\frac{a^{10}}{b^5}\).

Working with algebraic expressions involves recognizing patterns and applying known rules, such as the distributive property or combining like terms. These skills are fundamental to solving equations and inequalities, which is why gaining proficiency in handling algebraic expressions is critical for anyone studying mathematics or related fields.