Which Choice Is Equivalent To The Expression Below? ${ 2^{7.19} }$A. ${ 2^7 \cdot 2^{19 / 10} }$B. ${ 2^{7 + 1 / 10 + 9 / 10} }$C. ${ 2^7 \cdot 2^{1 / 10} \cdot 2^{9 / 100} }$D. $[ 2^7 + 2^{1 / 10} +

Introduction

Exponents are a fundamental concept in mathematics, used to represent repeated multiplication of a number. In this article, we will explore the concept of exponents and how to simplify expressions involving them. We will also examine the given expression and determine which choice is equivalent to it.

What are Exponents?

Exponents are a shorthand way of writing repeated multiplication of a number. For example, $2^3$ can be read as "2 to the power of 3" and is equivalent to $2 \times 2 \times 2$. Exponents can be used to represent any positive number raised to a power.

Properties of Exponents

There are several properties of exponents that are essential to understand when simplifying expressions. These properties include:

• Product of Powers: When multiplying two numbers with the same base, we can add their exponents. For example, $2^3 \times 2^4 = 2^{3+4} = 2^7$.
• Power of a Power: When raising a number with an exponent to another power, we can multiply the exponents. For example, $(2^3)^4 = 2^{3 \times 4} = 2^{12}$.
• Quotient of Powers: When dividing two numbers with the same base, we can subtract their exponents. For example, $\frac{2^5}{2^3} = 2^{5-3} = 2^2$.

Simplifying the Given Expression

The given expression is $2^{7.19}$. To simplify this expression, we can use the properties of exponents.

Step 1: Convert the Decimal to a Fraction

We can convert the decimal 7.19 to a fraction by writing it as $7 + 0.19$. We can then convert the decimal 0.19 to a fraction by writing it as $\frac{19}{100}$.

Step 2: Simplify the Expression

We can now simplify the expression by using the properties of exponents.

$2^{7.19} = 2^{7 + 0.19} = 2^{7 + \frac{19}{100}} = 2^7 \cdot 2^{\frac{19}{100}}$

Step 3: Simplify the Fractional Exponent

We can simplify the fractional exponent by writing it as a product of two fractions.

$2^{\frac{19}{100}} = 2^{\frac{10}{100} + \frac{9}{100}} = 2^{\frac{1}{10} + \frac{9}{100}}$

Step 4: Simplify the Expression Further

We can now simplify the expression further by combining the two terms.

$2^{7.19} = 2^7 \cdot 2^{\frac{1}{10} + \frac{9}{100}} = 2^7 \cdot 2^{\frac{1}{10}} \cdot 2^{\frac{9}{100}}$

Which Choice is Equivalent to the Expression?

Based on our simplification of the expression, we can see that the correct choice is:

C. $2^7 \cdot 2^{1 / 10} \cdot 2^{9 / 100}$

This choice is equivalent to the expression $2^{7.19}$.

Conclusion

In this article, we have explored the concept of exponents and how to simplify expressions involving them. We have also examined the given expression and determined which choice is equivalent to it. By understanding the properties of exponents and simplifying the expression step by step, we have arrived at the correct choice.

Key Takeaways

• Exponents are a fundamental concept in mathematics, used to represent repeated multiplication of a number.
• There are several properties of exponents that are essential to understand when simplifying expressions.
• By simplifying the given expression step by step, we can determine which choice is equivalent to it.

Further Reading

For further reading on exponents and simplifying expressions, we recommend the following resources:

• Khan Academy: Exponents and Exponential Functions
• Mathway: Exponents and Exponential Functions
• Wolfram MathWorld: Exponents and Exponential Functions
  Exponents and Equivalent Expressions: Q&A =============================================

Introduction

In our previous article, we explored the concept of exponents and how to simplify expressions involving them. We also examined the given expression and determined which choice is equivalent to it. In this article, we will answer some frequently asked questions related to exponents and equivalent expressions.

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

A: The base is the number being raised to a power, while the exponent is the power to which the base is being raised. For example, in the expression $2^3$, the base is 2 and the exponent is 3.

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

A: To simplify an expression with a fractional exponent, you can write it as a product of two fractions. For example, $2^{\frac{1}{2}} = 2^{\frac{1}{2} + \frac{0}{2}} = 2^{\frac{1}{2}} \cdot 2^{\frac{0}{2}} = \sqrt{2} \cdot 1 = \sqrt{2}$.

Q: Can I simplify an expression with a negative exponent?

A: Yes, you can simplify an expression with a negative exponent by using the property of exponents that states $a^{-n} = \frac{1}{a^n}$. For example, $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$.

Q: How do I simplify an expression with multiple bases?

A: To simplify an expression with multiple bases, you can use the property of exponents that states $a^m \cdot b^n = a^m \cdot b^n$. For example, $2^3 \cdot 3^2 = 2^3 \cdot 3^2$.

Q: Can I simplify an expression with a decimal exponent?

A: Yes, you can simplify an expression with a decimal exponent by converting the decimal to a fraction. For example, $2^{7.19} = 2^{7 + 0.19} = 2^{7 + \frac{19}{100}} = 2^7 \cdot 2^{\frac{19}{100}}$.

Q: How do I determine which choice is equivalent to an expression?

A: To determine which choice is equivalent to an expression, you can simplify the expression step by step and compare it to each choice. For example, in the expression $2^{7.19}$, we can simplify it to $2^7 \cdot 2^{\frac{1}{10} + \frac{9}{100}} = 2^7 \cdot 2^{\frac{1}{10}} \cdot 2^{\frac{9}{100}}$. We can then compare this expression to each choice and determine which one is equivalent to it.

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

A: Some common mistakes to avoid when simplifying expressions with exponents include:

• Not using the correct property of exponents
• Not simplifying the expression step by step
• Not converting decimals to fractions
• Not using the correct order of operations

Conclusion

In this article, we have answered some frequently asked questions related to exponents and equivalent expressions. We have also provided some tips and tricks for simplifying expressions with exponents. By understanding the properties of exponents and simplifying expressions step by step, you can determine which choice is equivalent to an expression.

Key Takeaways

• Exponents are a fundamental concept in mathematics, used to represent repeated multiplication of a number.
• There are several properties of exponents that are essential to understand when simplifying expressions.
• By simplifying the given expression step by step, you can determine which choice is equivalent to it.

Further Reading

For further reading on exponents and simplifying expressions, we recommend the following resources:

• Khan Academy: Exponents and Exponential Functions
• Mathway: Exponents and Exponential Functions
• Wolfram MathWorld: Exponents and Exponential Functions