Simplify The Following Expressions. Show All Calculations.1. { \sqrt{81} - \sqrt{49}$}$2. { \frac{\sqrt[3]{-125}}{-5^2}$}$3. { -6p^2(-p-3)$}$

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Introduction

Mathematical expressions can be complex and daunting, but with the right approach, they can be simplified to reveal their underlying structure. In this article, we will explore three mathematical expressions and demonstrate how to simplify them using step-by-step calculations.

Expression 1: Simplifying Square Roots

Problem Statement

Simplify the expression: 81−49\sqrt{81} - \sqrt{49}

Solution

To simplify this expression, we need to calculate the square roots of 81 and 49 separately.

81=9×9=9\sqrt{81} = \sqrt{9 \times 9} = 9

49=7×7=7\sqrt{49} = \sqrt{7 \times 7} = 7

Now, we can subtract the two square roots:

81−49=9−7=2\sqrt{81} - \sqrt{49} = 9 - 7 = 2

Therefore, the simplified expression is: 2\boxed{2}

Explanation

In this example, we used the property of square roots that states a×b=a×b\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}. We applied this property to simplify the square roots of 81 and 49, and then subtracted the two results to obtain the final answer.

Expression 2: Simplifying Cube Roots

Problem Statement

Simplify the expression: −1253−52\frac{\sqrt[3]{-125}}{-5^2}

Solution

To simplify this expression, we need to calculate the cube root of -125 and the square of -5 separately.

−1253=−5×−5×−53=−5\sqrt[3]{-125} = \sqrt[3]{-5 \times -5 \times -5} = -5

−52=−25-5^2 = -25

Now, we can divide the two results:

−1253−52=−5−25=15\frac{\sqrt[3]{-125}}{-5^2} = \frac{-5}{-25} = \frac{1}{5}

Therefore, the simplified expression is: 15\boxed{\frac{1}{5}}

Explanation

In this example, we used the property of cube roots that states a×b×c3=a3×b3×c3\sqrt[3]{a \times b \times c} = \sqrt[3]{a} \times \sqrt[3]{b} \times \sqrt[3]{c}. We applied this property to simplify the cube root of -125, and then divided the result by the square of -5 to obtain the final answer.

Expression 3: Simplifying Algebraic Expressions

Problem Statement

Simplify the expression: −6p2(−p−3)-6p^2(-p-3)

Solution

To simplify this expression, we need to apply the distributive property of multiplication over addition.

−6p2(−p−3)=−6p2×(−p)−6p2×3-6p^2(-p-3) = -6p^2 \times (-p) - 6p^2 \times 3

=6p3+18p2= 6p^3 + 18p^2

Therefore, the simplified expression is: 6p3+18p26p^3 + 18p^2

Explanation

In this example, we used the distributive property of multiplication over addition to simplify the expression. We applied this property to expand the product of -6p^2 and (-p-3), and then combined like terms to obtain the final answer.

Conclusion

Simplifying mathematical expressions is an essential skill in mathematics, and it requires a deep understanding of mathematical concepts and properties. In this article, we demonstrated how to simplify three mathematical expressions using step-by-step calculations. We used properties of square roots, cube roots, and algebraic expressions to simplify the expressions and obtain the final answers. By following these examples, you can develop your skills in simplifying mathematical expressions and tackle more complex problems with confidence.

Additional Tips and Resources

  • To simplify mathematical expressions, start by identifying the properties and rules that apply to the expression.
  • Use the distributive property of multiplication over addition to expand products.
  • Apply the properties of square roots and cube roots to simplify expressions involving these operations.
  • Combine like terms to simplify expressions and obtain the final answer.
  • Practice simplifying mathematical expressions to develop your skills and build confidence.

References

  • [1] "Algebra" by Michael Artin
  • [2] "Calculus" by Michael Spivak
  • [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Introduction

Simplifying mathematical expressions is an essential skill in mathematics, and it requires a deep understanding of mathematical concepts and properties. In this article, we will answer some frequently asked questions about simplifying mathematical expressions and provide additional tips and resources to help you develop your skills.

Q&A

Q: What is the difference between simplifying and solving a mathematical expression?

A: Simplifying a mathematical expression involves reducing it to its simplest form, while solving a mathematical expression involves finding the value of the expression. For example, simplifying the expression 2x+32x + 3 would result in 2x+32x + 3, while solving the expression 2x+3=52x + 3 = 5 would result in x=1x = 1.

Q: How do I simplify an expression with multiple operations?

A: To simplify an expression with multiple operations, start by identifying the operations and their order of precedence. For example, in the expression 3+2×43 + 2 \times 4, the multiplication operation takes precedence over the addition operation. Therefore, the expression would be simplified as follows:

3+2×4=3+8=113 + 2 \times 4 = 3 + 8 = 11

Q: How do I simplify an expression with fractions?

A: To simplify an expression with fractions, start by simplifying the fractions individually. For example, in the expression 23+14\frac{2}{3} + \frac{1}{4}, the fractions can be simplified as follows:

23=812\frac{2}{3} = \frac{8}{12}

14=312\frac{1}{4} = \frac{3}{12}

Now, the expression can be simplified as follows:

812+312=1112\frac{8}{12} + \frac{3}{12} = \frac{11}{12}

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, start by simplifying the exponents individually. For example, in the expression 23×222^3 \times 2^2, the exponents can be simplified as follows:

23=82^3 = 8

22=42^2 = 4

Now, the expression can be simplified as follows:

8×4=328 \times 4 = 32

Q: What are some common mistakes to avoid when simplifying mathematical expressions?

A: Some common mistakes to avoid when simplifying mathematical expressions include:

  • Not following the order of operations
  • Not simplifying fractions and exponents
  • Not combining like terms
  • Not checking for errors in calculations

Additional Tips and Resources

  • To simplify mathematical expressions, start by identifying the properties and rules that apply to the expression.
  • Use the distributive property of multiplication over addition to expand products.
  • Apply the properties of square roots and cube roots to simplify expressions involving these operations.
  • Combine like terms to simplify expressions and obtain the final answer.
  • Practice simplifying mathematical expressions to develop your skills and build confidence.
  • Use online resources, such as calculators and math software, to check your work and identify errors.

References

  • [1] "Algebra" by Michael Artin
  • [2] "Calculus" by Michael Spivak
  • [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Practice Problems

  • Simplify the expression 3x+2y−4z3x + 2y - 4z
  • Simplify the expression 12+13\frac{1}{2} + \frac{1}{3}
  • Simplify the expression 23×222^3 \times 2^2
  • Simplify the expression 16+9\sqrt{16} + \sqrt{9}

Answer Key

  • 3x+2y−4z=3x+2y−4z3x + 2y - 4z = 3x + 2y - 4z
  • 12+13=56\frac{1}{2} + \frac{1}{3} = \frac{5}{6}
  • 23×22=322^3 \times 2^2 = 32
  • 16+9=4+3=7\sqrt{16} + \sqrt{9} = 4 + 3 = 7