Evaluating Exponents and Fractions

Oct 13, 2025 by miodeya.com 15 views

Introduction

In mathematics, exponents and fractions are fundamental concepts that are used to represent and manipulate numbers. Exponents are used to represent repeated multiplication of a number, while fractions are used to represent a part of a whole. In this article, we will explore the concept of exponents and fractions, and how they are used to simplify expressions.

What are Exponents?

Exponents are a shorthand way of representing repeated multiplication of a number. For example, the expression $2^3$ represents the repeated multiplication of 2 by itself 3 times, which is equal to $2 \cdot 2 \cdot 2 = 8$ . Exponents are used to simplify expressions and make them easier to work with.

What are Fractions?

Fractions are a way of representing a part of a whole. A fraction is a ratio of two numbers, where the top number is called the numerator and the bottom number is called the denominator. For example, the fraction $\frac{1}{2}$ represents one half of a whole.

Evaluating Exponents and Fractions

Now that we have a basic understanding of exponents and fractions, let's evaluate the expression $\left(\frac{4}{5}\right)^6$ . To evaluate this expression, we need to apply the exponent rule, which states that $(a/b)^n = (a^n)/(b^n)$ .

Applying the Exponent Rule

Using the exponent rule, we can rewrite the expression $\left(\frac{4}{5}\right)^6$ as $\frac{4^6}{5^6}$ . This is because the exponent 6 is applied to both the numerator and the denominator.

Simplifying the Expression

Now that we have rewritten the expression, let's simplify it. To simplify the expression, we need to calculate the value of $4^6$ and $5^6$ . $4^6$ is equal to $4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 = 4096$ . $5^6$ is equal to $5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 = 15625$ .

Evaluating the Expression

Now that we have calculated the values of $4^6$ and $5^6$ , let's evaluate the expression $\frac{4^6}{5^6}$ . This is equal to $\frac{4096}{15625}$ .

Comparing the Options

Now that we have evaluated the expression, let's compare it to the options. Option A is $\frac{4^6}{5}$ . This is not equal to the expression we evaluated, because the denominator is not $5^6$ . Option B is $6 \cdot \left(\frac{4}{5}\right)$ . This is not equal to the expression we evaluated, because it does not have an exponent of 6. Option D is $\frac{24}{30}$ . This is not equal to the expression we evaluated, because it is not in the form of $\frac{a^6}{b^6}$ .

Conclusion

In conclusion, the correct answer is $\frac{4^6}{5^6}$ . This is because it is the only option that is equal to the expression $\left(\frac{4}{5}\right)^6$ . We used the exponent rule to rewrite the expression, and then simplified it by calculating the values of $4^6$ and $5^6$ . We compared the options and found that only $\frac{4^6}{5^6}$ is equal to the expression.

Final Answer

Q: What is the difference between an exponent and a power?

A: An exponent and a power are often used interchangeably, but technically, an exponent is the number that is being raised to a power, while a power is the result of raising a number to an exponent. For example, in the expression $2^3$ , 2 is the base and 3 is the exponent, while the result of raising 2 to the power of 3 is 8.

Q: How do I simplify an expression with an exponent?

A: To simplify an expression with an exponent, you need to apply the exponent rule, which states that $(a/b)^n = (a^n)/(b^n)$ . You can also use the rule that $a^m \cdot a^n = a^{m+n}$ to simplify expressions.

Q: What is the difference between a fraction and a decimal?

A: A fraction is a way of representing a part of a whole, while a decimal is a way of representing a number as a sum of powers of 10. For example, the fraction $\frac{1}{2}$ is equal to the decimal 0.5.

Q: How do I convert a fraction to a decimal?

A: To convert a fraction to a decimal, you can divide the numerator by the denominator. For example, to convert the fraction $\frac{1}{2}$ to a decimal, you can divide 1 by 2, which gives you 0.5.

Q: What is the difference between a rational number and an irrational number?

A: A rational number is a number that can be expressed as a fraction, while an irrational number is a number that cannot be expressed as a fraction. For example, the number 0.5 is a rational number, while the number $\sqrt{2}$ is an irrational number.

Q: How do I add and subtract fractions?

A: To add and subtract fractions, you need to have the same denominator. If the denominators are different, you need to find the least common multiple (LCM) of the denominators and then convert the fractions to have the LCM as the denominator.

Q: What is the difference between a positive exponent and a negative exponent?

A: A positive exponent is an exponent that is greater than 0, while a negative exponent is an exponent that is less than 0. For example, in the expression $2^3$ , the exponent 3 is positive, while in the expression $2^{-3}$ , the exponent -3 is negative.

Q: How do I evaluate an expression with a negative exponent?

A: To evaluate an expression with a negative exponent, you can use the rule that $a^{-n} = 1/a^n$ . For example, to evaluate the expression $2^{-3}$ , you can use the rule to get $1/2^3$ , which is equal to 1/8.

Q: What is the difference between a variable and a constant?

A: A variable is a symbol that represents a value that can change, while a constant is a value that does not change. For example, in the expression $x^2$ , x is a variable, while in the expression $2^3$ , 2 is a constant.

Q: How do I solve an equation with a variable?

A: To solve an equation with a variable, you need to isolate the variable on one side of the equation. You can do this by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation that can be written in the form $ax + b = c$ , where a, b, and c are constants, while a quadratic equation is an equation that can be written in the form $ax^2 + bx + c = 0$ , where a, b, and c are constants.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the quadratic formula, which is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . You can also use factoring to solve quadratic equations.

Conclusion

In conclusion, exponents and fractions are fundamental concepts in mathematics that are used to represent and manipulate numbers. Understanding these concepts is essential for solving equations and manipulating expressions. By following the rules and formulas outlined in this article, you can simplify expressions, add and subtract fractions, and solve equations with variables.