Problem 85 - Introductory and Intermediate Algebra for College Students 4th

Question

Will help you prepare for the material covered in the next section. 25 to what power gives \(5 ? \quad\left(25^{?}=5\right)\)

Step-by-Step Solution

Answer

\(x = 0.5\). So, 25 raised to the power of 0.5 gives 5.

Step 1: Rewrite the equation

Rewrite the equation with clear variable representation such as \(25^{x} = 5\).

Step 2: Rewrite 5 as base 25

To balance the equation, let's rewrite 5 as base 25. Knowing that 5 is the square root of 25, we can rewrite 5 as \(25^{0.5} (=25^{1/2})\). Our equation therefore becomes \(25^{x} = 25^{0.5}\).

Step 3: Conclude the equation

Using the property of exponents, if \(a^{m} = a^{n}\), then \(m=n\). Therefore, \(x = 0.5\).

Key Concepts

Rewriting Equations

Rewriting equations is a fundamental skill in algebra that aids in solving many mathematical problems. It's about expressing the same equation in a different form that makes the solution more evident or practical.
In the given exercise, the equation we start with is \(25^{?} = 5\). The challenge is to find what power of 25 results in 5.

• We can introduce a variable, let's call it \(x\), that represents the unknown power. This changes our original equation to \(25^{x} = 5\). Now, we have a clearer representation, and solving for \(x\) becomes our goal.
• Rewriting equations often involves recognizing patterns or employing algebraic identities, which simplify finding correct values.

Remember that rewriting equations is essentially about "speaking the same language" as the numbers and operations involved.

Properties of Exponents

Exponents are powerful tools in math, helping to simplify expressions and equations by using shorter, more manageable forms. There are certain properties of exponents that are very helpful.
Here are some key properties that are essential:

• **Product of Powers:** When multiplying like bases, add their exponents: \(a^m \cdot a^n = a^{m+n}\).
• **Quotient of Powers:** When dividing like bases, subtract the exponents: \(a^m / a^n = a^{m-n}\).
• **Power of a Power:** When raising an exponent to another power, multiply the exponents: \((a^m)^n = a^{m \cdot n}\).
• **Equal Base Rule:** If the bases are equal, the exponents must be equal: If \(a^m = a^n\), then \(m = n\).

In the exercise, one of the properties we used is the Equal Base Rule. We expressed both sides of the equation with the base 25 (i.e., \(25^x = 25^{0.5}\)), allowing us to equate \(x\) to 0.5.

Square Root as Exponent

When you see a square root, it can be rewritten in exponential form, which often simplifies the problem-solving process using properties of exponents. Understanding this concept allows for easier manipulation of an expression.

• For any number \(a\), the square root \(\sqrt{a}\) can be expressed as \(a^{0.5}\) or \(a^{1/2}\).
• This conversion is particularly useful when you're dealing with powers and roots because it brings them into the same 'language'.

In our problem, we had \(5\). Recognizing that 5 is the square root of 25 allowed us to rewrite 5 as \(25^{0.5}\). This conversion led to both sides of the equation having the same base, paving the way to apply the property of exponents to find the power of 0.5.