Simplify The Expression:$\[ 4^{-3} + 5(2 - 4) - 16^{3/4} \\]

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Introduction

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In this article, we will simplify the given mathematical expression step by step. The expression is $4^{-3} + 5(2 - 4) - 16^{3/4}$. We will use the order of operations (PEMDAS) to simplify the expression.

Understanding the Order of Operations

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The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The acronym PEMDAS stands for:

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

Step 1: Evaluate Expressions Inside Parentheses

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The first step is to evaluate the expression inside the parentheses: $2 - 4$. This expression is equal to $-2$.

Step 2: Evaluate Exponential Expressions

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Next, we need to evaluate the exponential expressions in the given expression. We have two exponential expressions: $4^{-3}$ and $16^{3/4}$.

Evaluating $4^{-3}$

To evaluate $4^{-3}$, we need to use the rule of negative exponents. A negative exponent means that we need to take the reciprocal of the base raised to the positive exponent. Therefore, $4^{-3}$ is equal to $\frac{1}{4^3}$.

Evaluating $16^{3/4}$

To evaluate $16^{3/4}$, we need to use the rule of fractional exponents. A fractional exponent means that we need to take the root of the base raised to the numerator and then raise the result to the power of the denominator. Therefore, $16^{3/4}$ is equal to $(16^{1/4})^3$.

Step 3: Simplify the Expression

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Now that we have evaluated the exponential expressions, we can simplify the given expression.

Simplifying $4^{-3}$

We have already simplified $4^{-3}$ to $\frac{1}{4^3}$.

Simplifying $16^{3/4}$

We have already simplified $16^{3/4}$ to $(16^{1/4})^3$.

Simplifying the Entire Expression

Now we can substitute the simplified expressions back into the original expression:

$\frac{1}{4^3} + 5(-2) - (16^{1/4})^3$

Step 4: Evaluate Multiplication and Division Operations

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Next, we need to evaluate the multiplication and division operations in the expression.

Evaluating $5(-2)$

The product of $5$ and $-2$ is $-10$.

Evaluating $(16^{1/4})^3$

To evaluate $(16^{1/4})^3$, we need to use the rule of fractional exponents. A fractional exponent means that we need to take the root of the base raised to the numerator and then raise the result to the power of the denominator. Therefore, $(16^{1/4})^3$ is equal to $16^{3/4}$.

Step 5: Simplify the Expression Further

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Now that we have evaluated the multiplication and division operations, we can simplify the expression further.

Simplifying the Entire Expression

Now we can substitute the evaluated expressions back into the original expression:

$\frac{1}{4^3} - 10 - 16^{3/4}$

Step 6: Evaluate Addition and Subtraction Operations

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Finally, we need to evaluate the addition and subtraction operations in the expression.

Evaluating $\frac{1}{4^3} - 10 - 16^{3/4}$

To evaluate $\frac{1}{4^3} - 10 - 16^{3/4}$, we need to follow the order of operations. First, we need to evaluate the exponential expressions. Then, we need to evaluate the multiplication and division operations. Finally, we need to evaluate the addition and subtraction operations.

Evaluating $\frac{1}{4^3}$

$\frac{1}{4^3}$ is equal to $\frac{1}{64}$.

Evaluating $16^{3/4}$

$16^{3/4}$ is equal to $8$.

Evaluating the Entire Expression

Now we can substitute the evaluated expressions back into the original expression:

$\frac{1}{64} - 10 - 8$

Step 7: Simplify the Final Expression

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Finally, we can simplify the final expression.

Simplifying the Final Expression

$\frac{1}{64} - 10 - 8$ is equal to $\frac{1}{64} - 18$.

Evaluating $\frac{1}{64} - 18$

To evaluate $\frac{1}{64} - 18$, we need to follow the order of operations. First, we need to evaluate the addition and subtraction operations. Therefore, $\frac{1}{64} - 18$ is equal to $-17.96875$.

The final answer is: $\boxed{-17.96875}$

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Introduction

---

In this article, we will simplify the given mathematical expression step by step. The expression is $4^{-3} + 5(2 - 4) - 16^{3/4}$. We will use the order of operations (PEMDAS) to simplify the expression.

Understanding the Order of Operations

---

The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The acronym PEMDAS stands for:

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

Step 1: Evaluate Expressions Inside Parentheses

---

The first step is to evaluate the expression inside the parentheses: $2 - 4$. This expression is equal to $-2$.

Step 2: Evaluate Exponential Expressions

---

Next, we need to evaluate the exponential expressions in the given expression. We have two exponential expressions: $4^{-3}$ and $16^{3/4}$.

Evaluating $4^{-3}$

To evaluate $4^{-3}$, we need to use the rule of negative exponents. A negative exponent means that we need to take the reciprocal of the base raised to the positive exponent. Therefore, $4^{-3}$ is equal to $\frac{1}{4^3}$.

Evaluating $16^{3/4}$

To evaluate $16^{3/4}$, we need to use the rule of fractional exponents. A fractional exponent means that we need to take the root of the base raised to the numerator and then raise the result to the power of the denominator. Therefore, $16^{3/4}$ is equal to $(16^{1/4})^3$.

Step 3: Simplify the Expression

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Now that we have evaluated the exponential expressions, we can simplify the given expression.

Simplifying $4^{-3}$

We have already simplified $4^{-3}$ to $\frac{1}{4^3}$.

Simplifying $16^{3/4}$

We have already simplified $16^{3/4}$ to $(16^{1/4})^3$.

Simplifying the Entire Expression

Now we can substitute the simplified expressions back into the original expression:

$\frac{1}{4^3} + 5(-2) - (16^{1/4})^3$

Step 4: Evaluate Multiplication and Division Operations

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Next, we need to evaluate the multiplication and division operations in the expression.

Evaluating $5(-2)$

The product of $5$ and $-2$ is $-10$.

Evaluating $(16^{1/4})^3$

To evaluate $(16^{1/4})^3$, we need to use the rule of fractional exponents. A fractional exponent means that we need to take the root of the base raised to the numerator and then raise the result to the power of the denominator. Therefore, $(16^{1/4})^3$ is equal to $16^{3/4}$.

Step 5: Simplify the Expression Further

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Now that we have evaluated the multiplication and division operations, we can simplify the expression further.

Simplifying the Entire Expression

Now we can substitute the evaluated expressions back into the original expression:

$\frac{1}{4^3} - 10 - 16^{3/4}$

Step 6: Evaluate Addition and Subtraction Operations

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Finally, we need to evaluate the addition and subtraction operations in the expression.

Evaluating $\frac{1}{4^3} - 10 - 16^{3/4}$

To evaluate $\frac{1}{4^3} - 10 - 16^{3/4}$, we need to follow the order of operations. First, we need to evaluate the exponential expressions. Then, we need to evaluate the multiplication and division operations. Finally, we need to evaluate the addition and subtraction operations.

Evaluating $\frac{1}{4^3}$

$\frac{1}{4^3}$ is equal to $\frac{1}{64}$.

Evaluating $16^{3/4}$

$16^{3/4}$ is equal to $8$.

Evaluating the Entire Expression

Now we can substitute the evaluated expressions back into the original expression:

$\frac{1}{64} - 10 - 8$

Step 7: Simplify the Final Expression

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Finally, we can simplify the final expression.

Simplifying the Final Expression

$\frac{1}{64} - 10 - 8$ is equal to $\frac{1}{64} - 18$.

Evaluating $\frac{1}{64} - 18$

To evaluate $\frac{1}{64} - 18$, we need to follow the order of operations. First, we need to evaluate the addition and subtraction operations. Therefore, $\frac{1}{64} - 18$ is equal to $-17.96875$.

The final answer is: $\boxed{-17.96875}$

Q&A

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Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The acronym PEMDAS stands for:

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

Q: How do I evaluate expressions inside parentheses?

A: To evaluate expressions inside parentheses, we need to follow the order of operations. First, we need to evaluate any exponential expressions. Then, we need to evaluate any multiplication and division operations. Finally, we need to evaluate any addition and subtraction operations.

Q: How do I evaluate exponential expressions?

A: To evaluate exponential expressions, we need to use the rules of exponents. A negative exponent means that we need to take the reciprocal of the base raised to the positive exponent. A fractional exponent means that we need to take the root of the base raised to the numerator and then raise the result to the power of the denominator.

Q: How do I simplify the expression $\frac{1}{4^3} + 5(-2) - (16^{1/4})^3$?

A: To simplify the expression $\frac{1}{4^3} + 5(-2) - (16^{1/4})^3$, we need to follow the order of operations. First, we need to evaluate the exponential expressions. Then, we need to evaluate the multiplication and division operations. Finally, we need to evaluate the addition and subtraction operations.

Q: What is the final answer to the expression $\frac{1}{4^3} + 5(-2) - (16^{1/4})^3$?

A: The final answer to the expression $\frac{1}{4^3} + 5(-2) - (16^{1/4})^3$ is $-17.96875$.