Which Expression Is Equivalent To − 32 3 5 -32^{\frac{3}{5}} − 3 2 5 3 ​ ?A. − 8 -8 − 8 B. − 32 5 3 -\sqrt[3]{32^5} − 3 3 2 5 ​ C. 1 32 5 3 \frac{1}{\sqrt[3]{32^5}} 3 3 2 5 ​ 1 ​ D. 1 8 \frac{1}{8} 8 1 ​

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Understanding the Problem

The given problem involves simplifying the expression $-32^{\frac{3}{5}}$. To solve this, we need to apply the rules of exponents and understand the properties of negative numbers. The expression $-32^{\frac{3}{5}}$ can be broken down into two parts: the negative sign and the exponent.

Breaking Down the Expression

The expression $-32^{\frac{3}{5}}$ can be rewritten as $-(32^{\frac{3}{5}})$. This is because the negative sign can be distributed to the entire expression inside the parentheses.

Applying the Rules of Exponents

To simplify the expression $32^{\frac{3}{5}}$, we need to apply the rule of exponents that states $(a^m)^n = a^{m \cdot n}$. In this case, we have $32^{\frac{3}{5}}$, which can be rewritten as $(32^{\frac{1}{5}})^3$.

Simplifying the Expression

Now, we need to simplify the expression $32^{\frac{1}{5}}$. To do this, we need to find the fifth root of 32. The fifth root of 32 is a number that, when raised to the power of 5, equals 32. This number is denoted by $\sqrt[5]{32}$.

Evaluating the Fifth Root of 32

The fifth root of 32 can be evaluated as $\sqrt[5]{32} = 2$. This is because $2^5 = 32$.

Simplifying the Expression Further

Now that we have simplified the expression $32^{\frac{1}{5}}$ to 2, we can substitute this value back into the original expression. We have $-(32^{\frac{3}{5}}) = -(2^3)$.

Evaluating the Expression

The expression $-(2^3)$ can be evaluated as $-8$. This is because $2^3 = 8$.

Conclusion

In conclusion, the expression $-32^{\frac{3}{5}}$ is equivalent to $-8$. This is because we can simplify the expression by applying the rules of exponents and evaluating the fifth root of 32.

Answer

The correct answer is A. $-8$.

Comparison with Other Options

Let's compare the correct answer with the other options.

• Option B: $-\sqrt[3]{32^5}$ is not equivalent to $-32^{\frac{3}{5}}$. This is because the cube root of $32^5$ is not equal to $32^{\frac{3}{5}}$.
• Option C: $\frac{1}{\sqrt[3]{32^5}}$ is not equivalent to $-32^{\frac{3}{5}}$. This is because the reciprocal of the cube root of $32^5$ is not equal to $-32^{\frac{3}{5}}$.
• Option D: $\frac{1}{8}$ is not equivalent to $-32^{\frac{3}{5}}$. This is because the reciprocal of 8 is not equal to $-32^{\frac{3}{5}}$.

Final Answer

The final answer is A. $-8$.

Key Takeaways

• The expression $-32^{\frac{3}{5}}$ can be simplified by applying the rules of exponents.
• The fifth root of 32 is 2.
• The expression $-32^{\frac{3}{5}}$ is equivalent to $-8$.

Common Mistakes

• Not applying the rules of exponents correctly.
• Not evaluating the fifth root of 32 correctly.
• Not simplifying the expression correctly.

Tips and Tricks

• Make sure to apply the rules of exponents correctly.
• Evaluate the fifth root of 32 correctly.
• Simplify the expression correctly.

Real-World Applications

• The concept of exponents and roots is used in various real-world applications, such as finance, science, and engineering.
• Understanding the properties of negative numbers and exponents is crucial in these applications.

Conclusion

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Frequently Asked Questions

Q: What is the difference between a positive and negative exponent?

A: A positive exponent indicates that the base is raised to a power, while a negative exponent indicates that the reciprocal of the base is raised to a power. For example, $a^m$ is different from $a^{-m}$.

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, you can rewrite it as the reciprocal of the base raised to the positive exponent. For example, $a^{-m}$ can be rewritten as $\frac{1}{a^m}$.

Q: What is the difference between a root and an exponent?

A: A root is the inverse operation of an exponent. For example, the square root of a number is the number that, when raised to the power of 2, equals the original number.

Q: How do I evaluate a root?

A: To evaluate a root, you need to find the number that, when raised to the power of the root, equals the original number. For example, to evaluate the square root of 16, you need to find the number that, when raised to the power of 2, equals 16.

Q: What is the difference between a rational and irrational root?

A: A rational root is a root that can be expressed as a fraction, while an irrational root is a root that cannot be expressed as a fraction.

Q: How do I simplify an expression with a rational root?

A: To simplify an expression with a rational root, you can rewrite it as the fraction of the base raised to the power of the root. For example, $\sqrt[3]{\frac{a}{b}}$ can be rewritten as $\frac{\sqrt[3]{a}}{\sqrt[3]{b}}$.

Q: What is the difference between a real and complex root?

A: A real root is a root that can be expressed as a real number, while a complex root is a root that cannot be expressed as a real number.

Q: How do I simplify an expression with a complex root?

A: To simplify an expression with a complex root, you need to use the properties of complex numbers. For example, to simplify the expression $\sqrt{-1}$, you need to use the property that $i^2 = -1$.

Q: What is the difference between a radical and a root?

A: A radical is a symbol that represents a root, while a root is the number that, when raised to the power of the radical, equals the original number.

Q: How do I simplify an expression with a radical?

A: To simplify an expression with a radical, you can rewrite it as the root of the base. For example, $\sqrt{a}$ can be rewritten as $a^{\frac{1}{2}}$.

Q: What is the difference between a perfect and imperfect square root?

A: A perfect square root is a root that can be expressed as a perfect square, while an imperfect square root is a root that cannot be expressed as a perfect square.

Q: How do I simplify an expression with a perfect square root?

A: To simplify an expression with a perfect square root, you can rewrite it as the square of the base. For example, $\sqrt{a^2}$ can be rewritten as $a$.

Q: What is the difference between a cube root and a square root?

A: A cube root is a root that represents the number that, when raised to the power of 3, equals the original number, while a square root is a root that represents the number that, when raised to the power of 2, equals the original number.

Q: How do I simplify an expression with a cube root?

A: To simplify an expression with a cube root, you can rewrite it as the cube of the base. For example, $\sqrt[3]{a^3}$ can be rewritten as $a$.

Q: What is the difference between a root and a logarithm?

A: A root is the inverse operation of an exponent, while a logarithm is the inverse operation of an exponent that represents the power to which a base must be raised to produce a given number.

Q: How do I simplify an expression with a logarithm?

A: To simplify an expression with a logarithm, you can rewrite it as the exponent of the base. For example, $\log_a{b}$ can be rewritten as $x$ if $a^x = b$.

Q: What is the difference between a natural and common logarithm?

A: A natural logarithm is a logarithm that has a base of $e$, while a common logarithm is a logarithm that has a base of 10.

Q: How do I simplify an expression with a natural logarithm?

A: To simplify an expression with a natural logarithm, you can rewrite it as the exponent of the base $e$. For example, $\ln{b}$ can be rewritten as $x$ if $e^x = b$.

Q: What is the difference between a logarithmic and exponential function?

A: A logarithmic function is a function that represents the inverse operation of an exponential function.

Q: How do I simplify an expression with a logarithmic function?

A: To simplify an expression with a logarithmic function, you can rewrite it as the exponent of the base. For example, $\log_a{b}$ can be rewritten as $x$ if $a^x = b$.

Q: What is the difference between a linear and nonlinear equation?

A: A linear equation is an equation that can be expressed in the form $ax + b = c$, while a nonlinear equation is an equation that cannot be expressed in this form.

Q: How do I simplify an expression with a linear equation?

A: To simplify an expression with a linear equation, you can rewrite it in the form $ax + b = c$ and solve for the variable.

Q: What is the difference between a quadratic and cubic equation?

A: A quadratic equation is an equation that can be expressed in the form $ax^2 + bx + c = 0$, while a cubic equation is an equation that can be expressed in the form $ax^3 + bx^2 + cx + d = 0$.

Q: How do I simplify an expression with a quadratic equation?

A: To simplify an expression with a quadratic equation, you can rewrite it in the form $ax^2 + bx + c = 0$ and solve for the variable.

Q: What is the difference between a rational and irrational number?

A: A rational number is a number that can be expressed as a fraction, while an irrational number is a number that cannot be expressed as a fraction.

Q: How do I simplify an expression with a rational number?

A: To simplify an expression with a rational number, you can rewrite it as the fraction of the base. For example, $\frac{a}{b}$ can be rewritten as $a \div b$.

Q: What is the difference between a real and complex number?

A: A real number is a number that can be expressed as a real number, while a complex number is a number that cannot be expressed as a real number.

Q: How do I simplify an expression with a complex number?

A: To simplify an expression with a complex number, you need to use the properties of complex numbers. For example, to simplify the expression $\sqrt{-1}$, you need to use the property that $i^2 = -1$.

Q: What is the difference between a polynomial and a rational function?

A: A polynomial is a function that can be expressed as a sum of terms, each of which is a power of the variable, while a rational function is a function that can be expressed as the ratio of two polynomials.

Q: How do I simplify an expression with a polynomial?

A: To simplify an expression with a polynomial, you can rewrite it as the sum of the terms. For example, $a^2 + 2ab + b^2$ can be rewritten as $(a + b)^2$.

Q: What is the difference between a rational and irrational function?

A: A rational function is a function that can be expressed as the ratio of two polynomials, while an irrational function is a function that cannot be expressed as the ratio of two polynomials.

Q: How do I simplify an expression with a rational function?

A: To simplify an expression with a rational function, you can rewrite it as the ratio of the two polynomials. For example, $\frac{a}{b}$ can be rewritten as $a \div b$.

Q: What is the difference between a linear and nonlinear inequality?

A: A linear inequality is an inequality that can be expressed in the form $ax + b < c$, while a nonlinear inequality is an inequality that cannot be expressed in this form.

Q: How do I simplify an expression with a linear inequality?

A: To simplify an expression with a linear inequality, you can rewrite it in the form $ax + b < c$ and solve for the variable.

Q: What is the difference between a quadratic and cubic inequality?

A: A quadratic inequality is an inequality that can be expressed in the form $ax^2 + bx + c < 0$, while a cubic inequality is an inequality that can be expressed in the form $ax^3 + bx^2 + cx + d < 0$.

Q: How do I simplify an expression with a quadratic inequality?

A: To simplify an expression with a quadratic inequality, you can rewrite it in the form $ax^2 + bx