Evaluate The Following Expressions Without The Use Of A Calculator:1. $8^{\frac{2}{3}}$2. $32^{-\frac{1}{5}}$3. $25^{-\frac{3}{2}}$4. $\left(\frac{9}{16}\right)^{\frac{3}{2}}$5.

Introduction

In mathematics, evaluating expressions without the use of a calculator is an essential skill that requires a deep understanding of mathematical concepts and operations. This article will guide you through the process of evaluating five different expressions without a calculator, focusing on the use of exponent rules, fraction operations, and mathematical properties.

Expression 1: $8^{\frac{2}{3}}$

To evaluate the expression $8^{\frac{2}{3}}$, we need to understand the concept of fractional exponents. A fractional exponent can be written as $a^{\frac{m}{n}}$, where $a$ is the base, $m$ is the numerator, and $n$ is the denominator. In this case, the base is $8$, the numerator is $2$, and the denominator is $3$.

We can rewrite the expression as $(8^{\frac{1}{3}})^2$. To evaluate this expression, we need to find the cube root of $8$ and then square the result.

The cube root of $8$ is $2$, since $2^3 = 8$. Therefore, $8^{\frac{1}{3}} = 2$. Now, we can square the result to get $(8^{\frac{1}{3}})^2 = 2^2 = 4$.

Expression 2: $32^{-\frac{1}{5}}$

To evaluate the expression $32^{-\frac{1}{5}}$, we need to understand the concept of negative exponents. A negative exponent can be written as $a^{-m}$, where $a$ is the base and $m$ is the exponent. In this case, the base is $32$ and the exponent is $-\frac{1}{5}$.

We can rewrite the expression as $\frac{1}{32^{\frac{1}{5}}}$. To evaluate this expression, we need to find the fifth root of $32$ and then take the reciprocal of the result.

The fifth root of $32$ is $2$, since $2^5 = 32$. Therefore, $32^{\frac{1}{5}} = 2$. Now, we can take the reciprocal of the result to get $\frac{1}{32^{\frac{1}{5}}} = \frac{1}{2} = 0.5$.

Expression 3: $25^{-\frac{3}{2}}$

To evaluate the expression $25^{-\frac{3}{2}}$, we need to understand the concept of negative exponents and fractional exponents. A negative exponent can be written as $a^{-m}$, where $a$ is the base and $m$ is the exponent. In this case, the base is $25$ and the exponent is $-\frac{3}{2}$.

We can rewrite the expression as $\frac{1}{25^{\frac{3}{2}}}$. To evaluate this expression, we need to find the square root of $25$ and then cube the result, and then take the reciprocal of the result.

The square root of $25$ is $5$, since $5^2 = 25$. Therefore, $25^{\frac{1}{2}} = 5$. Now, we can cube the result to get $5^3 = 125$. Finally, we can take the reciprocal of the result to get $\frac{1}{25^{\frac{3}{2}}} = \frac{1}{125} = 0.008$.

Expression 4: $\left(\frac{9}{16}\right)^{\frac{3}{2}}$

To evaluate the expression $\left(\frac{9}{16}\right)^{\frac{3}{2}}$, we need to understand the concept of fractional exponents and fraction operations. A fractional exponent can be written as $a^{\frac{m}{n}}$, where $a$ is the base, $m$ is the numerator, and $n$ is the denominator. In this case, the base is $\frac{9}{16}$, the numerator is $3$, and the denominator is $2$.

We can rewrite the expression as $\left(\frac{9}{16}\right)^{\frac{1}{2}} \cdot \left(\frac{9}{16}\right)^{\frac{2}{2}}$. To evaluate this expression, we need to find the square root of $\frac{9}{16}$ and then multiply the result by $\frac{9}{16}$.

The square root of $\frac{9}{16}$ is $\frac{3}{4}$, since $\left(\frac{3}{4}\right)^2 = \frac{9}{16}$. Therefore, $\left(\frac{9}{16}\right)^{\frac{1}{2}} = \frac{3}{4}$. Now, we can multiply the result by $\frac{9}{16}$ to get $\frac{3}{4} \cdot \frac{9}{16} = \frac{27}{64}$.

Expression 5: $\left(\frac{1}{8}\right)^{-\frac{3}{2}}$

To evaluate the expression $\left(\frac{1}{8}\right)^{-\frac{3}{2}}$, we need to understand the concept of negative exponents and fraction operations. A negative exponent can be written as $a^{-m}$, where $a$ is the base and $m$ is the exponent. In this case, the base is $\frac{1}{8}$ and the exponent is $-\frac{3}{2}$.

We can rewrite the expression as $\frac{1}{\left(\frac{1}{8}\right)^{\frac{3}{2}}}$. To evaluate this expression, we need to find the square root of $\frac{1}{8}$ and then cube the result, and then take the reciprocal of the result.

The square root of $\frac{1}{8}$ is $\frac{1}{2\sqrt{2}}$, since $\left(\frac{1}{2\sqrt{2}}\right)^2 = \frac{1}{8}$. Therefore, $\left(\frac{1}{8}\right)^{\frac{1}{2}} = \frac{1}{2\sqrt{2}}$. Now, we can cube the result to get $\left(\frac{1}{2\sqrt{2}}\right)^3 = \frac{1}{8\sqrt{2}}$. Finally, we can take the reciprocal of the result to get $\frac{1}{\left(\frac{1}{8}\right)^{\frac{3}{2}}} = \frac{1}{\frac{1}{8\sqrt{2}}} = 8\sqrt{2}$.

Conclusion

Introduction

In our previous article, we explored the process of evaluating expressions without a calculator, focusing on the use of exponent rules, fraction operations, and mathematical properties. In this article, we will address some common questions and concerns related to evaluating expressions without a calculator.

Q: What are some common mistakes to avoid when evaluating expressions without a calculator?

A: When evaluating expressions without a calculator, it's essential to avoid common mistakes such as:

• Not simplifying fractions before applying exponent rules
• Not following the order of operations (PEMDAS)
• Not using the correct exponent rules (e.g., negative exponents, fractional exponents)
• Not checking for errors in calculations

Q: How can I simplify fractions before applying exponent rules?

A: To simplify fractions before applying exponent rules, follow these steps:

• Factor the numerator and denominator into their prime factors
• Cancel out any common factors between the numerator and denominator
• Simplify the resulting fraction

For example, to simplify the fraction $\frac{12}{18}$, we can factor the numerator and denominator into their prime factors:

$\frac{12}{18} = \frac{2^2 \cdot 3}{2 \cdot 3^2}$

We can then cancel out the common factors between the numerator and denominator:

$\frac{2^2 \cdot 3}{2 \cdot 3^2} = \frac{2 \cdot 3}{3^2}$

Finally, we can simplify the resulting fraction:

$\frac{2 \cdot 3}{3^2} = \frac{2}{3}$

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: Evaluate expressions inside parentheses first
• Exponents: Evaluate any exponential expressions next
• Multiplication and Division: Evaluate any multiplication and division operations from left to right
• Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right

Q: How can I use exponent rules to evaluate expressions?

A: Exponent rules are a set of rules that dictate how to evaluate expressions with exponents. Some common exponent rules include:

• Product of Powers: When multiplying two powers with the same base, add the exponents
• Power of a Power: When raising a power to a power, multiply the exponents
• Negative Exponents: When a negative exponent is present, take the reciprocal of the base and change the sign of the exponent
• Fractional Exponents: When a fractional exponent is present, take the root of the base and raise it to the power of the numerator

For example, to evaluate the expression $2^3 \cdot 2^4$, we can use the product of powers rule:

$2^3 \cdot 2^4 = 2^{3+4} = 2^7$

Q: How can I check for errors in calculations?

A: To check for errors in calculations, follow these steps:

• Recheck your work: Go back and recheck your calculations to ensure that you made no mistakes
• Use a calculator: Use a calculator to verify your calculations
• Check for errors in exponent rules: Make sure that you applied the correct exponent rules
• Check for errors in fraction operations: Make sure that you simplified fractions correctly

Conclusion

Evaluating expressions without a calculator requires a deep understanding of mathematical concepts and operations. By following the rules of exponentiation, fraction operations, and mathematical properties, you can evaluate complex expressions and arrive at accurate results. In this article, we have addressed some common questions and concerns related to evaluating expressions without a calculator. By following these tips and understanding the underlying concepts, you can become proficient in evaluating expressions without a calculator.