Post Test: Scientific NotationSelect The Correct Answer.Simplify The Expression So There Is Only One Power For Each Base.$5.6^{-3} \cdot 3.4^{-7} \cdot 5.6^3 \cdot 3.4^{-4}$A. $5.6^{-2} \cdot 3.4^{-11}$ B. $5.6^8 \cdot

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Understanding Scientific Notation

Scientific notation is a way of expressing very large or very small numbers in a more manageable form. It consists of a number between 1 and 10, multiplied by a power of 10. For example, the number 456,000 can be written in scientific notation as 4.56 × 10^5. Scientific notation is commonly used in mathematics, physics, and engineering to simplify complex calculations.

Simplifying Expressions with Exponents

When simplifying expressions with exponents, we need to apply the rules of exponentiation. One of the key rules is that when multiplying two numbers with the same base, we add their exponents. For example, (2^3) × (2^4) = 2^(3+4) = 2^7. On the other hand, when dividing two numbers with the same base, we subtract their exponents. For example, (2^3) ÷ (2^2) = 2^(3-2) = 2^1.

Simplifying the Given Expression

The given expression is 5.6−3⋅3.4−7⋅5.63⋅3.4−45.6^{-3} \cdot 3.4^{-7} \cdot 5.6^3 \cdot 3.4^{-4}. To simplify this expression, we need to apply the rules of exponentiation. We can start by combining the terms with the same base.

Combining Terms with the Same Base

We can combine the terms with the base 5.6 as follows:

5.6−3⋅5.63=5.6(−3)+3=5.60=15.6^{-3} \cdot 5.6^3 = 5.6^{(-3)+3} = 5.6^0 = 1

Similarly, we can combine the terms with the base 3.4 as follows:

3.4−7⋅3.4−4=3.4(−7)+(−4)=3.4−113.4^{-7} \cdot 3.4^{-4} = 3.4^{(-7)+(-4)} = 3.4^{-11}

Simplifying the Expression

Now that we have combined the terms with the same base, we can simplify the expression as follows:

5.6−3⋅3.4−7⋅5.63⋅3.4−4=1⋅3.4−11=3.4−115.6^{-3} \cdot 3.4^{-7} \cdot 5.6^3 \cdot 3.4^{-4} = 1 \cdot 3.4^{-11} = 3.4^{-11}

Answer

The correct answer is 3.4−113.4^{-11}.

Explanation

The correct answer is 3.4−113.4^{-11} because we combined the terms with the same base and applied the rules of exponentiation. We can rewrite the expression as 5.6−3⋅5.63⋅3.4−7⋅3.4−4=1⋅3.4−11=3.4−115.6^{-3} \cdot 5.6^3 \cdot 3.4^{-7} \cdot 3.4^{-4} = 1 \cdot 3.4^{-11} = 3.4^{-11}.

Conclusion

In conclusion, when simplifying expressions with exponents, we need to apply the rules of exponentiation. We can combine the terms with the same base and add or subtract their exponents. In this case, we combined the terms with the base 5.6 and 3.4 and applied the rules of exponentiation to simplify the expression.

Common Mistakes to Avoid

When simplifying expressions with exponents, there are several common mistakes to avoid. One of the common mistakes is to forget to combine the terms with the same base. Another common mistake is to apply the wrong rule of exponentiation. For example, when multiplying two numbers with the same base, we add their exponents, but when dividing two numbers with the same base, we subtract their exponents.

Tips and Tricks

When simplifying expressions with exponents, there are several tips and tricks to keep in mind. One of the tips is to always combine the terms with the same base. Another tip is to apply the rules of exponentiation carefully. For example, when multiplying two numbers with the same base, we add their exponents, but when dividing two numbers with the same base, we subtract their exponents.

Practice Problems

When simplifying expressions with exponents, it is essential to practice problems to reinforce your understanding. Here are a few practice problems to try:

  1. Simplify the expression 23⋅24⋅2−22^3 \cdot 2^4 \cdot 2^{-2}.
  2. Simplify the expression 35⋅3−3⋅323^5 \cdot 3^{-3} \cdot 3^2.
  3. Simplify the expression 42⋅4−3⋅444^2 \cdot 4^{-3} \cdot 4^4.

Answer Key

  1. 2(3+4)+(−2)=252^{(3+4)+(-2)} = 2^5
  2. 3(5+(−3))+2=343^{(5+(-3))+2} = 3^4
  3. 4(2+(−3))+4=434^{(2+(-3))+4} = 4^3

Conclusion

Frequently Asked Questions

Q: What is scientific notation?

A: Scientific notation is a way of expressing very large or very small numbers in a more manageable form. It consists of a number between 1 and 10, multiplied by a power of 10.

Q: How do I simplify expressions with exponents?

A: To simplify expressions with exponents, you need to apply the rules of exponentiation. When multiplying two numbers with the same base, you add their exponents. When dividing two numbers with the same base, you subtract their exponents.

Q: What is the rule for combining terms with the same base?

A: When combining terms with the same base, you add their exponents. For example, 23⋅24=2(3+4)=272^3 \cdot 2^4 = 2^{(3+4)} = 2^7.

Q: What is the rule for dividing terms with the same base?

A: When dividing terms with the same base, you subtract their exponents. For example, 23÷22=2(3−2)=212^3 \div 2^2 = 2^{(3-2)} = 2^1.

Q: How do I simplify the expression 5.6−3⋅3.4−7⋅5.63⋅3.4−45.6^{-3} \cdot 3.4^{-7} \cdot 5.6^3 \cdot 3.4^{-4}?

A: To simplify the expression, you need to combine the terms with the same base and apply the rules of exponentiation. You can start by combining the terms with the base 5.6 and 3.4.

Q: What is the correct answer for the expression 5.6−3⋅3.4−7⋅5.63⋅3.4−45.6^{-3} \cdot 3.4^{-7} \cdot 5.6^3 \cdot 3.4^{-4}?

A: The correct answer is 3.4−113.4^{-11}.

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

A: Some common mistakes to avoid include forgetting to combine the terms with the same base and applying the wrong rule of exponentiation.

Q: What are some tips and tricks for simplifying expressions with exponents?

A: Some tips and tricks include always combining the terms with the same base and applying the rules of exponentiation carefully.

Q: How can I practice simplifying expressions with exponents?

A: You can practice simplifying expressions with exponents by trying out different problems and applying the rules of exponentiation.

Q: What are some examples of expressions with exponents that I can practice simplifying?

A: Some examples of expressions with exponents that you can practice simplifying include:

  • 23⋅24⋅2−22^3 \cdot 2^4 \cdot 2^{-2}
  • 35⋅3−3⋅323^5 \cdot 3^{-3} \cdot 3^2
  • 42⋅4−3⋅444^2 \cdot 4^{-3} \cdot 4^4

Q: What is the answer key for the practice problems?

A: The answer key for the practice problems is:

  • 23⋅24⋅2−2=252^3 \cdot 2^4 \cdot 2^{-2} = 2^5
  • 35⋅3−3⋅32=343^5 \cdot 3^{-3} \cdot 3^2 = 3^4
  • 42⋅4−3⋅44=434^2 \cdot 4^{-3} \cdot 4^4 = 4^3

Conclusion

In conclusion, simplifying expressions with exponents requires careful application of the rules of exponentiation. By practicing problems and applying the rules of exponentiation carefully, you can simplify expressions with exponents with confidence.