Complete The Parts Below To Define $7^{\frac{1}{3}}$.(a) Let's See What Happens When We Raise $7^{\frac{1}{3}}$ To An Exponent Of 3. Use The Power Of A Power Rule To Fill In The Blanks.$\left(7 {\frac{1}{3}}\right) 3 =

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Unlocking the Secrets of Exponents: A Comprehensive Guide to Evaluating $7^{\frac{1}{3}}$

Exponents are a fundamental concept in mathematics, and understanding how to work with them is crucial for solving a wide range of mathematical problems. In this article, we will delve into the world of exponents and explore how to evaluate the expression $7^{\frac{1}{3}}$. We will use the power of a power rule to simplify this expression and gain a deeper understanding of exponents.

What are Exponents?

Exponents are a shorthand way of representing repeated multiplication. For example, $2^3$ can be read as "2 to the power of 3" or "2 cubed". This means that we need to multiply 2 by itself 3 times: $2^3 = 2 \times 2 \times 2 = 8$. Exponents can be positive or negative, and they can also be fractions or decimals.

The Power of a Power Rule

The power of a power rule is a fundamental concept in exponentiation. It states that when we raise a power to another power, we can multiply the exponents. In other words, $(am)n = a^{m \times n}$. This rule allows us to simplify complex expressions involving exponents.

Evaluating $7^{\frac{1}{3}}$

Now that we have a solid understanding of exponents and the power of a power rule, let's apply this knowledge to evaluate the expression $7^{\frac{1}{3}}$. We are asked to raise $7^{\frac{1}{3}}$ to an exponent of 3 and use the power of a power rule to fill in the blanks.

(713)3=?\left(7^{\frac{1}{3}}\right)^3 = ?

Using the power of a power rule, we can rewrite the expression as:

(713)3=713×3\left(7^{\frac{1}{3}}\right)^3 = 7^{\frac{1}{3} \times 3}

Now, we can simplify the exponent by multiplying $\frac{1}{3}$ by 3:

713×3=717^{\frac{1}{3} \times 3} = 7^1

Finally, we can evaluate the expression by raising 7 to the power of 1:

71=77^1 = 7

Therefore, $\left(7{\frac{1}{3}}\right)3 = 7$.

In this article, we have explored the concept of exponents and the power of a power rule. We have used this knowledge to evaluate the expression $7^{\frac{1}{3}}$ and gain a deeper understanding of exponents. By applying the power of a power rule, we were able to simplify the expression and arrive at the final answer.

  • Q: What is the power of a power rule? A: The power of a power rule states that when we raise a power to another power, we can multiply the exponents.
  • Q: How do I evaluate the expression $7^\frac{1}{3}}$? A To evaluate the expression $7^{\frac{1{3}}$, we need to raise it to an exponent of 3 and use the power of a power rule to simplify the expression.
  • Q: What is the final answer to the expression $\left(7{\frac{1}{3}}\right)3$? A: The final answer to the expression $\left(7{\frac{1}{3}}\right)3$ is 7.
  • Khan Academy: Exponents and Exponential Functions
  • Mathway: Exponents and Exponential Functions
  • Wolfram Alpha: Exponents and Exponential Functions
  • "Algebra and Trigonometry" by Michael Sullivan
  • "Calculus" by Michael Spivak
  • "Mathematics for the Nonmathematician" by Morris Kline
    Exponents and Exponential Functions: A Q&A Guide =====================================================

Exponents and exponential functions are fundamental concepts in mathematics, and understanding them is crucial for solving a wide range of mathematical problems. In this article, we will provide a comprehensive Q&A guide to help you better understand exponents and exponential functions.

Q: What is an exponent?

A: An exponent is a shorthand way of representing repeated multiplication. For example, $2^3$ can be read as "2 to the power of 3" or "2 cubed". This means that we need to multiply 2 by itself 3 times: $2^3 = 2 \times 2 \times 2 = 8$.

Q: What is the power of a power rule?

A: The power of a power rule is a fundamental concept in exponentiation. It states that when we raise a power to another power, we can multiply the exponents. In other words, $(am)n = a^{m \times n}$.

Q: How do I evaluate an expression with exponents?

A: To evaluate an expression with exponents, you need to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate any expressions inside parentheses first.
  2. Exponents: Evaluate any exponents next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

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

A: An exponential function is a function that can be written in the form $f(x) = a^x$, where $a$ is a positive constant. A polynomial function, on the other hand, is a function that can be written in the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Q: How do I graph an exponential function?

A: To graph an exponential function, you can use the following steps:

  1. Determine the base of the exponential function. This is the value of $a$ in the equation $f(x) = a^x$.
  2. Determine the horizontal asymptote of the function. This is the value that the function approaches as $x$ approaches infinity.
  3. Plot a few points on the graph to get an idea of the shape of the function.
  4. Use a graphing calculator or software to plot the function and get a more accurate graph.

Q: What is the domain and range of an exponential function?

A: The domain of an exponential function is all real numbers, since any real number can be used as the input $x$. The range of an exponential function is all positive real numbers, since the output $f(x)$ is always positive.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you can use the following steps:

  1. Isolate the exponential term on one side of the equation.
  2. Use the fact that $a^x = a^y$ implies $x = y$ to solve for the variable.
  3. Check your solution by plugging it back into the original equation.

Q: What is the difference between an exponential growth and an exponential decay?

A: Exponential growth occurs when a quantity increases at a rate that is proportional to the current value of the quantity. Exponential decay occurs when a quantity decreases at a rate that is proportional to the current value of the quantity.

In this article, we have provided a comprehensive Q&A guide to help you better understand exponents and exponential functions. We have covered topics such as the power of a power rule, evaluating expressions with exponents, graphing exponential functions, and solving exponential equations. We hope that this guide has been helpful in your understanding of exponents and exponential functions.

  • Q: What is the power of a power rule? A: The power of a power rule states that when we raise a power to another power, we can multiply the exponents.
  • Q: How do I evaluate an expression with exponents? A: To evaluate an expression with exponents, you need to follow the order of operations (PEMDAS).
  • Q: What is the difference between an exponential function and a polynomial function? A: An exponential function is a function that can be written in the form $f(x) = a^x$, where $a$ is a positive constant. A polynomial function, on the other hand, is a function that can be written in the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.
  • Khan Academy: Exponents and Exponential Functions
  • Mathway: Exponents and Exponential Functions
  • Wolfram Alpha: Exponents and Exponential Functions
  • "Algebra and Trigonometry" by Michael Sullivan
  • "Calculus" by Michael Spivak
  • "Mathematics for the Nonmathematician" by Morris Kline