Simplify The Expression:$\[ \frac{-3^3}{9} + \frac{8^2}{(-4)^2} \\]

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Introduction

Mathematical expressions can be complex and daunting, but with a clear understanding of the rules and operations involved, they can be simplified and evaluated with ease. In this article, we will focus on simplifying the given expression: −339+82(−4)2\frac{-3^3}{9} + \frac{8^2}{(-4)^2}. We will break down the expression into smaller parts, apply the rules of arithmetic operations, and finally arrive at the simplified result.

Understanding the Expression

The given expression is a combination of two fractions: −339\frac{-3^3}{9} and 82(−4)2\frac{8^2}{(-4)^2}. To simplify this expression, we need to evaluate each fraction separately and then add them together.

Evaluating the First Fraction

The first fraction is −339\frac{-3^3}{9}. To evaluate this fraction, we need to follow the order of operations (PEMDAS):

  1. Evaluate the exponent: −33=−27-3^3 = -27
  2. Divide the result by 9: −279=−3\frac{-27}{9} = -3

So, the first fraction simplifies to −3-3.

Evaluating the Second Fraction

The second fraction is 82(−4)2\frac{8^2}{(-4)^2}. To evaluate this fraction, we need to follow the order of operations (PEMDAS):

  1. Evaluate the exponents: 82=648^2 = 64 and (−4)2=16(-4)^2 = 16
  2. Divide the result by 16: 6416=4\frac{64}{16} = 4

So, the second fraction simplifies to 44.

Adding the Fractions

Now that we have simplified both fractions, we can add them together:

−339+82(−4)2=−3+4=1\frac{-3^3}{9} + \frac{8^2}{(-4)^2} = -3 + 4 = 1

Therefore, the simplified expression is 11.

Conclusion

Simplifying mathematical expressions requires a clear understanding of the rules and operations involved. By breaking down the expression into smaller parts, applying the rules of arithmetic operations, and finally adding the fractions together, we can arrive at the simplified result. In this article, we simplified the expression −339+82(−4)2\frac{-3^3}{9} + \frac{8^2}{(-4)^2} and arrived at the final result of 11.

Frequently Asked Questions

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

A: The order of operations (PEMDAS) is a set of rules that dictate the order in which mathematical operations should be performed. The acronym PEMDAS stands for Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.

Q: How do I evaluate exponents?

A: To evaluate exponents, you need to raise the base number to the power of the exponent. For example, 23=2×2×2=82^3 = 2 \times 2 \times 2 = 8.

Q: How do I add fractions?

A: To add 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 each fraction to have the LCM as the denominator.

Final Answer

The final answer is 1\boxed{1}.

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Introduction

Mathematical expressions can be complex and daunting, but with a clear understanding of the rules and operations involved, they can be simplified and evaluated with ease. In this article, we will focus on simplifying the given expression: −339+82(−4)2\frac{-3^3}{9} + \frac{8^2}{(-4)^2}. We will break down the expression into smaller parts, apply the rules of arithmetic operations, and finally arrive at the simplified result.

Understanding the Expression

The given expression is a combination of two fractions: −339\frac{-3^3}{9} and 82(−4)2\frac{8^2}{(-4)^2}. To simplify this expression, we need to evaluate each fraction separately and then add them together.

Evaluating the First Fraction

The first fraction is −339\frac{-3^3}{9}. To evaluate this fraction, we need to follow the order of operations (PEMDAS):

  1. Evaluate the exponent: −33=−27-3^3 = -27
  2. Divide the result by 9: −279=−3\frac{-27}{9} = -3

So, the first fraction simplifies to −3-3.

Evaluating the Second Fraction

The second fraction is 82(−4)2\frac{8^2}{(-4)^2}. To evaluate this fraction, we need to follow the order of operations (PEMDAS):

  1. Evaluate the exponents: 82=648^2 = 64 and (−4)2=16(-4)^2 = 16
  2. Divide the result by 16: 6416=4\frac{64}{16} = 4

So, the second fraction simplifies to 44.

Adding the Fractions

Now that we have simplified both fractions, we can add them together:

−339+82(−4)2=−3+4=1\frac{-3^3}{9} + \frac{8^2}{(-4)^2} = -3 + 4 = 1

Therefore, the simplified expression is 11.

Conclusion

Simplifying mathematical expressions requires a clear understanding of the rules and operations involved. By breaking down the expression into smaller parts, applying the rules of arithmetic operations, and finally adding the fractions together, we can arrive at the simplified result. In this article, we simplified the expression −339+82(−4)2\frac{-3^3}{9} + \frac{8^2}{(-4)^2} and arrived at the final result of 11.

Frequently Asked Questions

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

A: The order of operations (PEMDAS) is a set of rules that dictate the order in which mathematical operations should be performed. The acronym PEMDAS stands for Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.

Q: How do I evaluate exponents?

A: To evaluate exponents, you need to raise the base number to the power of the exponent. For example, 23=2×2×2=82^3 = 2 \times 2 \times 2 = 8.

Q: How do I add fractions?

A: To add 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 each fraction to have the LCM as the denominator.

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

A: A fraction is a way of expressing a part of a whole, while a decimal is a way of expressing a number as a sum of powers of 10. For example, the fraction 12\frac{1}{2} is equal to the decimal 0.5.

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator and the denominator and then divide both numbers by the GCD.

Q: What is the least common multiple (LCM)?

A: The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers. For example, the LCM of 4 and 6 is 12.

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

A: To convert a fraction to a decimal, you need to divide the numerator by the denominator. For example, the fraction 12\frac{1}{2} is equal to the decimal 0.5.

Final Answer

The final answer is 1\boxed{1}.

Additional Resources

Conclusion

Simplifying mathematical expressions requires a clear understanding of the rules and operations involved. By breaking down the expression into smaller parts, applying the rules of arithmetic operations, and finally adding the fractions together, we can arrive at the simplified result. In this article, we simplified the expression −339+82(−4)2\frac{-3^3}{9} + \frac{8^2}{(-4)^2} and arrived at the final result of 11. We also provided additional resources for further learning and practice.