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Chapter 7: Problem 21

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

Short Answer

Expert verified
The expression \(\frac{3^0}{4^0}\) evaluates to 1.

Step by step solution

01

Identify the Problem

The exercise involves evaluating the expression \( \frac{3^0}{4^0} \) when \( x eq 0 \). This step confirms that since there is no \( x \) explicitly in the expression, the condition \( x eq 0 \) does not directly affect this expression.
02

Evaluate Exponents

Start by recalling the rule for any non-zero number raised to the power of 0, which states that \( a^0 = 1 \). This rule applies because any number to the power of zero is considered to be 1 by definition.
03

Simplify the Expression

Using the rule from Step 2, convert both numerators and denominators to 1: \( 3^0 = 1 \) and \( 4^0 = 1 \). Therefore, the expression becomes \( \frac{1}{1} \).
04

Compute the Final Result

Since \(\frac{1}{1}\) simplifies to 1, this is our final result of the expression.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rational Expressions
A rational expression is a fraction where both the numerator and the denominator are polynomials. It's similar to fractions in arithmetic, but instead of numbers, you're dealing with algebraic expressions.
An example is \( \frac{x+2}{y-3} \), where \( x+2 \) is the numerator and \( y-3 \) is the denominator.
When working with rational expressions:
  • Simplify by factoring both the numerator and the denominator.
  • Cancel common factors between the numerator and the denominator.
  • Remember to consider restrictions, which are values that make the denominator zero, as they are not allowed.
Understanding rational expressions is crucial for solving various types of algebra problems, including those involving exponents, equations, and inequalities.
Simplifying Expressions
Simplifying an expression means reducing it to its simplest form, where it cannot be simplified further. This often involves combining like terms, factoring, and using mathematical identities.
For example, if you encounter \( \frac{3^0}{4^0} \):
  • Recall that \( 3^0 = 1 \) and \( 4^0 = 1 \) due to the zero exponent rule.
  • The expression then becomes \( \frac{1}{1} \), which simplifies to 1.
So, by applying the zero exponent rule, we simplify complex expressions easily. When simplified, expressions are typically easier to understand and work with in subsequent calculations.
Basic Exponents
Exponents are a fundamental component of algebra. They indicate how many times a number, known as the base, is multiplied by itself.
Starting with the basics:
  • \( a^n \) means the base \( a \) is used as a factor \( n \) times.
  • When \( n = 0 \), by definition, \( a^0 = 1 \) for any non-zero \( a \). This is known as the zero exponent rule.
Understanding basic exponents is critical as they are foundational in manipulating and simplifying more complex algebraic expressions. When learning about exponents, keep these tips in mind:
  • Practice the power rules: \((a^m)^n = a^{mn}\) and \(a^m \times a^n = a^{m+n}\).
  • Recognize special cases like \(a^0 = 1\) to simplify expressions quickly.
Mastering these basics will aid you in tackling more advanced problems, including those involving rational expressions.

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Most popular questions from this chapter

In \(64-75,\) write each quotient as a product without a denominator. The variables are not equal to zero. $$ \frac{20 x^{0} y^{-5}}{4 x^{-1} y^{5}} $$

In \(38-57,\) write each radical expression as a power with positive exponents and express the answer in simplest form. The variables are positive numbers. $$ \sqrt[4]{3} $$

In \(35-63,\) write each expression with only positive exponents and express the answer in simplest form. The variables are not equal to zero. $$ \frac{(-x)^{-5}}{x^{-3}} $$

In \(11-16,\) use the formula \(A=A_{0}\left(1+\frac{r}{n}\right)^{n t}\) to find the missing variable to the nearest hundredth. $$ A=100, A_{0}=25, n=1, t=2 $$

In \(58-73\) , write each power as a radical expression in simplest form. The variables are positive numbers. $$ \left(16 a^{5} b^{6}\right)^{\frac{1}{4}} $$
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