4. \[$\left(\frac{3}{5}\right)^2\$\]A. \[$\frac{3}{5} \times 2 = \frac{6}{5}\$\]B. \[$\frac{3}{5} \times \frac{3}{5} = \frac{9}{25}\$\]C. \[$\frac{3}{5} \times 2 = \frac{5}{7}\$\]D. \[$\frac{3}{5} \times \frac{3}{5}

by ADMIN 216 views

Introduction

In mathematics, exponents and fractions are two fundamental concepts that are often used together. Exponents represent repeated multiplication of a number, while fractions represent a part of a whole. In this article, we will explore how to solve exponents and multiply fractions, using a specific example to illustrate the process.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication of a number. For example, the expression (35)2\left(\frac{3}{5}\right)^2 represents the product of 35\frac{3}{5} multiplied by itself. To evaluate this expression, we need to multiply the fraction by itself.

Multiplying Fractions

Multiplying fractions involves multiplying the numerators (the numbers on top) and the denominators (the numbers on the bottom) separately. For example, to multiply 35\frac{3}{5} by 35\frac{3}{5}, we multiply the numerators (3 and 3) and the denominators (5 and 5) separately.

Solving the Given Expression

Now, let's apply the concepts of exponents and multiplying fractions to the given expression: (35)2\left(\frac{3}{5}\right)^2. To solve this expression, we need to multiply the fraction by itself.

(35)2=35×35\left(\frac{3}{5}\right)^2 = \frac{3}{5} \times \frac{3}{5}

Using the rule for multiplying fractions, we multiply the numerators (3 and 3) and the denominators (5 and 5) separately.

35×35=3×35×5\frac{3}{5} \times \frac{3}{5} = \frac{3 \times 3}{5 \times 5}

Now, we simplify the expression by multiplying the numbers in the numerator and the denominator.

3×35×5=925\frac{3 \times 3}{5 \times 5} = \frac{9}{25}

Therefore, the solution to the given expression is 925\frac{9}{25}.

Comparing the Solution to the Options

Now, let's compare the solution we obtained to the options provided.

A. 35×2=65\frac{3}{5} \times 2 = \frac{6}{5}

This option is incorrect because it multiplies the fraction by 2, rather than multiplying it by itself.

B. 35×35=925\frac{3}{5} \times \frac{3}{5} = \frac{9}{25}

This option is correct because it multiplies the fraction by itself, resulting in the same solution we obtained.

C. 35×2=57\frac{3}{5} \times 2 = \frac{5}{7}

This option is incorrect because it multiplies the fraction by 2, rather than multiplying it by itself, and also results in a different solution.

D. 35×35\frac{3}{5} \times \frac{3}{5}

This option is incomplete because it does not provide the solution to the expression.

Conclusion

In conclusion, solving exponents and multiplying fractions involves applying the rules for exponents and multiplying fractions. By following these rules, we can simplify expressions and obtain the correct solution. In this article, we used the expression (35)2\left(\frac{3}{5}\right)^2 to illustrate the process of solving exponents and multiplying fractions. We compared the solution we obtained to the options provided and determined that option B was the correct solution.

Frequently Asked Questions

Q: What is the rule for multiplying fractions?

A: The rule for multiplying fractions involves multiplying the numerators (the numbers on top) and the denominators (the numbers on the bottom) separately.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you need to multiply the fraction by itself. For example, to simplify the expression (35)2\left(\frac{3}{5}\right)^2, you need to multiply the fraction by itself.

Q: What is the difference between multiplying fractions and multiplying decimals?

A: Multiplying fractions involves multiplying the numerators and denominators separately, while multiplying decimals involves multiplying the numbers as is.

Q: Can I use a calculator to solve expressions with exponents?

A: Yes, you can use a calculator to solve expressions with exponents. However, it's always a good idea to understand the underlying math concepts and rules to ensure accuracy.

Additional Resources

For more information on solving exponents and multiplying fractions, check out the following resources:

  • Khan Academy: Exponents and Fractions
  • Mathway: Exponents and Fractions
  • Wolfram Alpha: Exponents and Fractions

Final Thoughts

Q: What is the rule for multiplying fractions?

A: The rule for multiplying fractions involves multiplying the numerators (the numbers on top) and the denominators (the numbers on the bottom) separately. For example, to multiply 35\frac{3}{5} by 35\frac{3}{5}, you multiply the numerators (3 and 3) and the denominators (5 and 5) separately.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you need to multiply the fraction by itself. For example, to simplify the expression (35)2\left(\frac{3}{5}\right)^2, you need to multiply the fraction by itself.

Q: What is the difference between multiplying fractions and multiplying decimals?

A: Multiplying fractions involves multiplying the numerators and denominators separately, while multiplying decimals involves multiplying the numbers as is. For example, to multiply 0.5 by 0.5, you multiply the numbers as is, resulting in 0.25.

Q: Can I use a calculator to solve expressions with exponents?

A: Yes, you can use a calculator to solve expressions with exponents. However, it's always a good idea to understand the underlying math concepts and rules to ensure accuracy.

Q: How do I handle negative exponents?

A: To handle negative exponents, you need to take the reciprocal of the fraction and change the sign of the exponent. For example, to simplify the expression (35)−2\left(\frac{3}{5}\right)^{-2}, you take the reciprocal of the fraction and change the sign of the exponent, resulting in (53)2\left(\frac{5}{3}\right)^2.

Q: Can I simplify expressions with multiple exponents?

A: Yes, you can simplify expressions with multiple exponents by applying the rules for exponents. For example, to simplify the expression (35)2×(35)3\left(\frac{3}{5}\right)^2 \times \left(\frac{3}{5}\right)^3, you apply the rules for exponents, resulting in (35)5\left(\frac{3}{5}\right)^5.

Q: How do I handle expressions with variables in the exponent?

A: To handle expressions with variables in the exponent, you need to apply the rules for exponents and variables. For example, to simplify the expression (2x)2\left(2x\right)^2, you apply the rules for exponents and variables, resulting in 4x24x^2.

Q: Can I use exponents to simplify expressions with fractions?

A: Yes, you can use exponents to simplify expressions with fractions. For example, to simplify the expression 35×35\frac{3}{5} \times \frac{3}{5}, you can use exponents to simplify the expression, resulting in (35)2\left(\frac{3}{5}\right)^2.

Q: How do I handle expressions with fractions and decimals?

A: To handle expressions with fractions and decimals, you need to convert the decimals to fractions and then apply the rules for fractions. For example, to simplify the expression 0.5×0.50.5 \times 0.5, you convert the decimals to fractions, resulting in 12×12\frac{1}{2} \times \frac{1}{2}, and then apply the rules for fractions, resulting in 14\frac{1}{4}.

Q: Can I use a calculator to convert fractions to decimals?

A: Yes, you can use a calculator to convert fractions to decimals. However, it's always a good idea to understand the underlying math concepts and rules to ensure accuracy.

Q: How do I handle expressions with negative fractions?

A: To handle expressions with negative fractions, you need to apply the rules for negative fractions. For example, to simplify the expression −35-\frac{3}{5}, you apply the rules for negative fractions, resulting in −35-\frac{3}{5}.

Q: Can I use exponents to simplify expressions with negative fractions?

A: Yes, you can use exponents to simplify expressions with negative fractions. For example, to simplify the expression −(35)2-\left(\frac{3}{5}\right)^2, you apply the rules for exponents and negative fractions, resulting in −925-\frac{9}{25}.

Conclusion

In conclusion, exponents and fractions are fundamental concepts in mathematics that are used to simplify expressions and solve problems. By understanding the rules and concepts, you can simplify expressions and obtain the correct solution. Remember to always follow the rules for exponents and fractions, and don't be afraid to ask for help if you need it.

Additional Resources

For more information on exponents and fractions, check out the following resources:

  • Khan Academy: Exponents and Fractions
  • Mathway: Exponents and Fractions
  • Wolfram Alpha: Exponents and Fractions

Final Thoughts

Exponents and fractions are essential skills in mathematics that are used to simplify expressions and solve problems. By understanding the rules and concepts, you can simplify expressions and obtain the correct solution. Remember to always follow the rules for exponents and fractions, and don't be afraid to ask for help if you need it.