Problem 19 - Algebra 2 and Trigonometry

Question

In \(11-22,\) find the value of each expression when \(x \neq 0\) $$ \left(\frac{3}{4}\right)^{0} $$

Step-by-Step Solution

Answer

The value is 1.

Step 1: Recognize the Zero Exponent Rule

The zero exponent rule states that any non-zero number raised to the power of zero is equal to 1. This is a fundamental rule in exponents that applies regardless of the base of the number.

Step 2: Apply the Zero Exponent Rule

For the expression \( \left(\frac{3}{4}\right)^{0} \), apply the zero exponent rule. Since the base \( \frac{3}{4} eq 0 \), the expression evaluates to 1.

Key Concepts

Understanding Exponents

Exponents are a way to express repeated multiplication of a number by itself. If you have a number, say 3, and you see it raised to the power of 2, written as \(3^2\), it means \(3 \times 3 = 9\).
This is read as "3 squared" and is a simple way to handle multiplication of a number by itself multiple times.Here are some key points about exponents:

• An exponent tells you how many times to use the base in a multiplication.
• Exponents are written as small numbers to the top right of a base number.
• The base is the number that is being multiplied.

Understanding how exponents work can simplify more complex mathematics, as it compresses long multiplication into a manageable form.

Exploring Algebra Basics

Algebra is like a language made up of symbols, often involving letters to represent numbers. This allows for a universal way to express mathematical ideas and relationships. The most fundamental concept of algebra is the equation, which shows that two expressions are equal.Consider a simple equation:\[x + 3 = 7\]Here, \(x\) is an unknown value that you need to find. In algebra, you'll learn various techniques to solve for unknowns, making algebra a powerful tool.Some basic principles in algebra include:

• Maintaining balance: You have to do the same operation on both sides of an equation.
• Combining like terms: This simplifies equations by merging similar variables.
• Using inverse operations: These are operations that undo each other, like addition and subtraction.

Algebra helps in systematically thinking through problems and finding unknown values.

Evaluating Expressions

Evaluating expressions is all about finding the value of an expression by substituting numbers for variables and carrying out arithmetic operations. Whether you're simplifying or substituting values, understanding the order of operations is essential.Let's take an example of evaluating a simple expression:Given \(2x + 3\) when \(x = 5\), substitute 5 for \(x\):\[2(5) + 3 = 10 + 3 = 13\]The expression evaluates to 13 when \(x = 5\).Some important tips for evaluating expressions:

• Substitute numbers for all variables present in the expression.
• Always follow the order of operations: Parentheses first, then exponents, followed by multiplication or division (from left to right), and finally, addition or subtraction.
• Simplify the entire expression step by step to get to the final value.

Being thorough and understanding every part of the expression ensures accurate evaluation.