Find The Value Of Each Of The Following Expressions.a) $\sqrt{8^4}$b) $\sqrt[3]{-343}$

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Introduction

In mathematics, expressions are a combination of numbers, variables, and mathematical operations. Evaluating expressions involves simplifying them to find their value. In this article, we will focus on finding the value of two given expressions, which involve square roots and cube roots. We will use the properties of exponents and roots to simplify these expressions.

Evaluating Expression a) 84\sqrt{8^4}

To evaluate the expression 84\sqrt{8^4}, we need to follow the order of operations (PEMDAS):

  1. Evaluate the exponent: 84=8×8×8×8=40968^4 = 8 \times 8 \times 8 \times 8 = 4096
  2. Take the square root of the result: 4096=64\sqrt{4096} = 64

Therefore, the value of the expression 84\sqrt{8^4} is 64.

Evaluating Expression b) −3433\sqrt[3]{-343}

To evaluate the expression −3433\sqrt[3]{-343}, we need to follow the order of operations (PEMDAS):

  1. Evaluate the cube root: −3433=−7\sqrt[3]{-343} = -7

Therefore, the value of the expression −3433\sqrt[3]{-343} is -7.

Understanding the Properties of Exponents and Roots

In the previous section, we used the properties of exponents and roots to simplify the expressions. Let's take a closer look at these properties:

  • Exponent Properties: When we have an expression with an exponent, such as aba^b, we can simplify it by evaluating the exponent first. For example, 23=2×2×2=82^3 = 2 \times 2 \times 2 = 8.
  • Root Properties: When we have an expression with a root, such as a\sqrt{a}, we can simplify it by evaluating the root first. For example, 16=4\sqrt{16} = 4.

Using the Properties of Exponents and Roots to Simplify Expressions

Now that we have a good understanding of the properties of exponents and roots, let's use them to simplify some expressions:

  • Example 1: 162\sqrt{16^2}
    • Evaluate the exponent: 162=16×16=25616^2 = 16 \times 16 = 256
    • Take the square root of the result: 256=16\sqrt{256} = 16
  • Example 2: −2723\sqrt[3]{-27^2}
    • Evaluate the exponent: −272=(−27)×(−27)=729-27^2 = (-27) \times (-27) = 729
    • Take the cube root of the result: 7293=−9\sqrt[3]{729} = -9

Conclusion

In this article, we evaluated two expressions involving square roots and cube roots. We used the properties of exponents and roots to simplify these expressions and find their value. We also discussed the importance of following the order of operations (PEMDAS) when evaluating expressions. By understanding the properties of exponents and roots, we can simplify complex expressions and find their value.

Frequently Asked Questions

  • Q: What is the value of 84\sqrt{8^4}?
    • A: The value of 84\sqrt{8^4} is 64.
  • Q: What is the value of −3433\sqrt[3]{-343}?
    • A: The value of −3433\sqrt[3]{-343} is -7.
  • Q: How do I simplify an expression with an exponent?
    • A: To simplify an expression with an exponent, evaluate the exponent first and then simplify the resulting expression.
  • Q: How do I simplify an expression with a root?
    • A: To simplify an expression with a root, evaluate the root first and then simplify the resulting expression.

Final Thoughts

Evaluating expressions is an essential skill in mathematics. By understanding the properties of exponents and roots, we can simplify complex expressions and find their value. In this article, we evaluated two expressions involving square roots and cube roots and used the properties of exponents and roots to simplify them. We also discussed the importance of following the order of operations (PEMDAS) when evaluating expressions. By following these steps, we can simplify any expression and find its value.

Introduction

In our previous article, we discussed how to evaluate expressions involving square roots and cube roots. We also covered the properties of exponents and roots and how to simplify complex expressions. In this article, we will answer some frequently asked questions related to evaluating expressions.

Q&A

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when evaluating an expression. The acronym PEMDAS stands for:

  • P: Parentheses (evaluate expressions inside parentheses first)
  • E: Exponents (evaluate exponents next)
  • M: Multiplication and Division (evaluate multiplication and division operations from left to right)
  • A: Addition and Subtraction (evaluate addition and subtraction operations from left to right)

Q: How do I evaluate an expression with multiple exponents?

A: To evaluate an expression with multiple exponents, follow the order of operations (PEMDAS). First, evaluate the exponents from left to right. For example:

  • 84×23\sqrt{8^4 \times 2^3}
    • Evaluate the exponents: 84=40968^4 = 4096 and 23=82^3 = 8
    • Multiply the results: 4096×8=327684096 \times 8 = 32768
    • Take the square root of the result: 32768=181\sqrt{32768} = 181

Q: How do I evaluate an expression with multiple roots?

A: To evaluate an expression with multiple roots, follow the order of operations (PEMDAS). First, evaluate the roots from left to right. For example:

  • −343×233\sqrt[3]{-343 \times 2^3}
    • Evaluate the exponents: 23=82^3 = 8
    • Multiply the results: −343×8=−2744-343 \times 8 = -2744
    • Take the cube root of the result: −27443=−14\sqrt[3]{-2744} = -14

Q: What is the difference between a square root and a cube root?

A: A square root is a root that is raised to the power of 1/2, while a cube root is a root that is raised to the power of 1/3. For example:

  • 16=4\sqrt{16} = 4 (square root)
  • −273=−3\sqrt[3]{-27} = -3 (cube root)

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

A: To simplify an expression with a negative exponent, follow the rule:

  • a−n=1ana^{-n} = \frac{1}{a^n}

For example:

  • 2−3=123=182^{-3} = \frac{1}{2^3} = \frac{1}{8}

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

A: To simplify an expression with a fractional exponent, follow the rule:

  • am/n=amna^{m/n} = \sqrt[n]{a^m}

For example:

  • 23/4=234=84=22^{3/4} = \sqrt[4]{2^3} = \sqrt[4]{8} = 2

Conclusion

In this article, we answered some frequently asked questions related to evaluating expressions. We covered topics such as the order of operations (PEMDAS), evaluating expressions with multiple exponents, and simplifying expressions with negative and fractional exponents. By following these rules and guidelines, we can simplify complex expressions and find their value.

Final Thoughts

Evaluating expressions is an essential skill in mathematics. By understanding the properties of exponents and roots, we can simplify complex expressions and find their value. In this article, we answered some frequently asked questions related to evaluating expressions and provided examples to illustrate the concepts. We hope that this article has been helpful in clarifying any doubts you may have had about evaluating expressions.

Additional Resources

  • Mathematics Handbook: A comprehensive guide to mathematics, including topics such as algebra, geometry, and calculus.
  • Mathematics Online Resources: A collection of online resources, including tutorials, videos, and practice problems, to help you learn and practice mathematics.
  • Mathematics Software: A list of software programs, including calculators and computer algebra systems, to help you evaluate expressions and solve mathematical problems.

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