Which Expressions Have A Value Between 0 And 1? Select All That Apply.A. $\left(\frac{4}{5}\right)^{-3}$B. $5^{-7} \cdot 4^2$C. $\frac{5^{-3}}{4^6}$D. $\frac{4^{-2}}{5^{-3}}$E. $4^{-3} \cdot 5^{-9}$

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Which Expressions Have a Value Between 0 and 1? Select All That Apply

In mathematics, understanding the properties of exponents and fractions is crucial for solving various problems. One of the key concepts is identifying expressions that have a value between 0 and 1. This article will explore five different expressions and determine which ones satisfy this condition.

Understanding the Basics of Exponents and Fractions

Before diving into the expressions, let's review the basics of exponents and fractions. Exponents represent the power to which a number is raised. For example, aba^b means aa multiplied by itself bb times. Fractions, on the other hand, represent a part of a whole. They consist of a numerator and a denominator, separated by a division symbol.

Expression A: (45)−3\left(\frac{4}{5}\right)^{-3}

To evaluate this expression, we need to apply the rule of negative exponents, which states that a−n=1ana^{-n} = \frac{1}{a^n}. Applying this rule, we get:

(45)−3=1(45)3=14353=5343=12564\left(\frac{4}{5}\right)^{-3} = \frac{1}{\left(\frac{4}{5}\right)^3} = \frac{1}{\frac{4^3}{5^3}} = \frac{5^3}{4^3} = \frac{125}{64}

Since 12564\frac{125}{64} is greater than 1, expression A does not have a value between 0 and 1.

Expression B: 5−7⋅425^{-7} \cdot 4^2

To evaluate this expression, we need to apply the rule of negative exponents and the rule of multiplying powers with the same base. Applying these rules, we get:

5−7⋅42=157⋅42=4257=16781255^{-7} \cdot 4^2 = \frac{1}{5^7} \cdot 4^2 = \frac{4^2}{5^7} = \frac{16}{78125}

Since 1678125\frac{16}{78125} is less than 1, expression B has a value between 0 and 1.

Expression C: 5−346\frac{5^{-3}}{4^6}

To evaluate this expression, we need to apply the rule of negative exponents and the rule of dividing powers with the same base. Applying these rules, we get:

5−346=15346=153⋅146=153⋅14096=178125⋅4096\frac{5^{-3}}{4^6} = \frac{\frac{1}{5^3}}{4^6} = \frac{1}{5^3} \cdot \frac{1}{4^6} = \frac{1}{5^3} \cdot \frac{1}{4096} = \frac{1}{78125 \cdot 4096}

Since 178125â‹…4096\frac{1}{78125 \cdot 4096} is less than 1, expression C has a value between 0 and 1.

Expression D: 4−25−3\frac{4^{-2}}{5^{-3}}

To evaluate this expression, we need to apply the rule of negative exponents and the rule of dividing powers with the same base. Applying these rules, we get:

4−25−3=142153=142⋅531=12516\frac{4^{-2}}{5^{-3}} = \frac{\frac{1}{4^2}}{\frac{1}{5^3}} = \frac{1}{4^2} \cdot \frac{5^3}{1} = \frac{125}{16}

Since 12516\frac{125}{16} is greater than 1, expression D does not have a value between 0 and 1.

Expression E: 4−3⋅5−94^{-3} \cdot 5^{-9}

To evaluate this expression, we need to apply the rule of negative exponents and the rule of multiplying powers with the same base. Applying these rules, we get:

4−3⋅5−9=143⋅159=164⋅11953125=11250000004^{-3} \cdot 5^{-9} = \frac{1}{4^3} \cdot \frac{1}{5^9} = \frac{1}{64} \cdot \frac{1}{1953125} = \frac{1}{125000000}

Since 1125000000\frac{1}{125000000} is less than 1, expression E has a value between 0 and 1.

Conclusion

In conclusion, expressions B, C, and E have a value between 0 and 1. Expression A does not have a value between 0 and 1, and expression D has a value greater than 1. Understanding the properties of exponents and fractions is crucial for solving various problems in mathematics. By applying the rules of exponents and fractions, we can determine which expressions satisfy the condition of having a value between 0 and 1.

Key Takeaways

  • Exponents represent the power to which a number is raised.
  • Fractions represent a part of a whole.
  • The rule of negative exponents states that a−n=1ana^{-n} = \frac{1}{a^n}.
  • The rule of multiplying powers with the same base states that amâ‹…an=am+na^m \cdot a^n = a^{m+n}.
  • The rule of dividing powers with the same base states that aman=am−n\frac{a^m}{a^n} = a^{m-n}.

Final Answer

The final answer is:

  • Expression A: No
  • Expression B: Yes
  • Expression C: Yes
  • Expression D: No
  • Expression E: Yes
    Q&A: Understanding Expressions with Values Between 0 and 1

In our previous article, we explored five different expressions and determined which ones have a value between 0 and 1. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the condition for an expression to have a value between 0 and 1?

A: An expression has a value between 0 and 1 if it is a fraction with a numerator less than the denominator, or if it is a negative number.

Q: How do I determine if an expression has a value between 0 and 1?

A: To determine if an expression has a value between 0 and 1, you need to evaluate the expression and check if it is a fraction with a numerator less than the denominator, or if it is a negative number.

Q: What is the rule of negative exponents?

A: The rule of negative exponents states that a−n=1ana^{-n} = \frac{1}{a^n}. This means that a negative exponent can be rewritten as a fraction with a numerator of 1 and a denominator of ana^n.

Q: How do I apply the rule of negative exponents?

A: To apply the rule of negative exponents, you need to rewrite the expression with a negative exponent as a fraction with a numerator of 1 and a denominator of ana^n. For example, a−n=1ana^{-n} = \frac{1}{a^n}.

Q: What is the rule of multiplying powers with the same base?

A: The rule of multiplying powers with the same base states that amâ‹…an=am+na^m \cdot a^n = a^{m+n}. This means that when you multiply two powers with the same base, you add the exponents.

Q: How do I apply the rule of multiplying powers with the same base?

A: To apply the rule of multiplying powers with the same base, you need to add the exponents. For example, amâ‹…an=am+na^m \cdot a^n = a^{m+n}.

Q: What is the rule of dividing powers with the same base?

A: The rule of dividing powers with the same base states that aman=am−n\frac{a^m}{a^n} = a^{m-n}. This means that when you divide two powers with the same base, you subtract the exponents.

Q: How do I apply the rule of dividing powers with the same base?

A: To apply the rule of dividing powers with the same base, you need to subtract the exponents. For example, aman=am−n\frac{a^m}{a^n} = a^{m-n}.

Q: Can you give an example of an expression with a value between 0 and 1?

A: Yes, an example of an expression with a value between 0 and 1 is 12\frac{1}{2}. This expression has a numerator of 1 and a denominator of 2, which is less than 1.

Q: Can you give an example of an expression with a value greater than 1?

A: Yes, an example of an expression with a value greater than 1 is 21\frac{2}{1}. This expression has a numerator of 2 and a denominator of 1, which is greater than 1.

Q: Can you give an example of an expression with a value less than 0?

A: Yes, an example of an expression with a value less than 0 is −1-1. This expression is a negative number, which is less than 0.

Conclusion

In conclusion, understanding expressions with values between 0 and 1 requires a good grasp of the rules of exponents and fractions. By applying these rules, you can determine which expressions satisfy the condition of having a value between 0 and 1. We hope this Q&A article has been helpful in answering your questions and providing a better understanding of this topic.

Key Takeaways

  • An expression has a value between 0 and 1 if it is a fraction with a numerator less than the denominator, or if it is a negative number.
  • The rule of negative exponents states that a−n=1ana^{-n} = \frac{1}{a^n}.
  • The rule of multiplying powers with the same base states that amâ‹…an=am+na^m \cdot a^n = a^{m+n}.
  • The rule of dividing powers with the same base states that aman=am−n\frac{a^m}{a^n} = a^{m-n}.

Final Answer

The final answer is:

  • Expression A: No
  • Expression B: Yes
  • Expression C: Yes
  • Expression D: No
  • Expression E: Yes