Problem 14 - Algebra 1: Concepts and Skills

Question

Copy and complete the statement. \(5^{?} \cdot 5^{8}=5^{9}\)

Step-by-Step Solution

Answer

The unknown exponent (?) equals 1.

Step 1: Identify the Equation

First identify the equation that needs to be solved, which is \(5^{?} \cdot 5^{8}=5^{9}\). Here, the question mark (?) is the unknown exponent we need to find.

Step 2: Apply Exponent Addition Rule

The multiplication \(5^{?} \cdot 5^{8}\) can be rewritten according to the exponent rule that states when multiplying two expressions with the same base, we add the exponents. This gives \(5^{?+8}\). Therefore, the equation becomes \(5^{?+8} =5^{9}\).

Step 3: Write an Equation to Solve for ?

By observing the equation \(5^{?+8} =5^{9}\), we can equate the exponents because the bases are the same. Therefore, \(?+8 =9\).

Step 4: Solve for ?

Solve for the unknown by subtracting 8 from both sides of the equation. This gives \(?=9-8\)

Step 5: Final Result

Compute the operation \(9-8\) to find the value of the unknown, which equals to 1.

Key Concepts

Understanding Exponents

Exponents are a way to express repeated multiplication of the same number. For example, the expression \(5^3\) means \(5 \times 5 \times 5\). The small number, known as the exponent, shows how many times to multiply the base number by itself.

Working with exponents can make calculations simpler, especially with large numbers and complex equations. They often appear in algebra, physics, and computer science.

Here are some basic rules to remember about exponents:

• **Product of Powers Rule:** When multiplying two powers with the same base, add their exponents (e.g., \(a^m \cdot a^n = a^{m+n}\)).
• **Quotient of Powers Rule:** When dividing two powers with the same base, subtract their exponents (e.g., \(a^m / a^n = a^{m-n}\)).
• **Power of a Power Rule:** When raising a power to another power, multiply the exponents (e.g., \((a^m)^n = a^{m \cdot n}\)).

Understanding these basic rules will help you tackle algebraic expressions more effectively.

Multiplication of Powers

When you multiply powers with the same base, you apply one of the core exponent rules: the product of powers rule. This rule states that if you have the same base, you add the exponents together.

For example:

• \(5^2 \cdot 5^3 = 5^{2+3} = 5^5\)
• Adding the exponents \(2 + 3\) simplifies the multiplication to \(5^5\).

This rule is essential in simplifying expressions and solving equations involving exponents.

When you encounter problems involving the multiplication of powers, it's always a good first step to check if the bases are the same. If they are, simply add up the exponents to obtain your solution. This not only simplifies the math but also helps in quickly finding answers without needing lengthy calculations.

Solving Algebraic Equations with Exponents

Algebraic equations can seem challenging, but solving them is much easier with a good grasp of exponent rules. When you solve equations involving exponents, you're often required to isolate the variable or unknown.

Consider an example problem:
The equation \(5^{?} \cdot 5^{8}=5^{9}\) asks us to find the missing exponent, represented by \(?\).

Here's a step-by-step approach:

• Use the product of powers rule to express the multiplication as a single exponent: \(5^{?+8}\).
• Set this equal to the other side of the equation: \(5^{?+8} = 5^9\).
• With matching bases, equate the exponents: \(?+8 = 9\).
• Solve for the unknown by subtracting 8 from both sides: \(? = 9 - 8\).
• Simplify to find \(? = 1\).

By breaking down each step and applying these techniques, you can successfully handle algebraic equations and uncover the unknown value with ease.