Type The Correct Answer In The Box. Use Numerals Instead Of Words.Consider This Expression: A 3 − 7 + ∣ B ∣ \sqrt{a^3-7} + |b| A 3 − 7 ​ + ∣ B ∣ When A = 2 A=2 A = 2 And B = − 4 B=-4 B = − 4 , The Value Of The Expression Is ___________.

Consider this expression: $\sqrt{a^3-7} + |b|$

When $a=2$ and $b=-4$, the value of the expression is ___________.

Step 1: Evaluate the expression inside the square root

To find the value of the expression, we first need to evaluate the expression inside the square root. The expression inside the square root is $a^3-7$. We are given that $a=2$, so we substitute $a=2$ into the expression.

Step 2: Substitute $a=2$ into the expression

Substituting $a=2$ into the expression $a^3-7$, we get $2^3-7$. Evaluating this expression, we get $8-7=1$.

Step 3: Evaluate the absolute value of $b$

Next, we need to evaluate the absolute value of $b$. We are given that $b=-4$, so we substitute $b=-4$ into the absolute value expression.

Step 4: Substitute $b=-4$ into the absolute value expression

Substituting $b=-4$ into the absolute value expression $|b|$, we get $|-4|$. Evaluating this expression, we get $4$.

Step 5: Add the results of the square root and absolute value expressions

Now that we have evaluated the expressions inside the square root and absolute value, we can add the results together. The expression inside the square root evaluated to $1$, and the absolute value expression evaluated to $4$. Adding these results together, we get $1+4=5$.

Step 6: Write the final answer

Therefore, the value of the expression is $\boxed{5}$.

Conclusion

In this problem, we were given an expression $\sqrt{a^3-7} + |b|$ and asked to find its value when $a=2$ and $b=-4$. We evaluated the expressions inside the square root and absolute value, and then added the results together to get the final answer.

Key Takeaways

• When evaluating expressions with square roots and absolute values, it's essential to follow the order of operations (PEMDAS).
• Substituting given values into expressions can help simplify the problem and make it easier to solve.
• Adding the results of evaluated expressions can give us the final answer.

Related Problems

• Evaluate the expression $\sqrt{a^2-4} + |b|$ when $a=3$ and $b=-2$.
• Find the value of the expression $\sqrt{b^2-9} + |a|$ when $a=1$ and $b=4$.

Final Answer

The final answer is: $\boxed{5}$

Introduction

In this article, we will answer some frequently asked questions about evaluating expressions with square roots and absolute values. These types of expressions can be challenging to work with, but with the right techniques and strategies, you can master them.

Q1: What is the difference between a square root and an absolute value?

A1: A square root is a mathematical operation that finds the number that, when multiplied by itself, gives a specified value. An absolute value, on the other hand, is the distance of a number from zero on the number line, without considering direction.

Q2: How do I evaluate an expression with a square root and an absolute value?

A2: To evaluate an expression with a square root and an absolute value, you need to follow the order of operations (PEMDAS). First, evaluate the expression inside the square root, then evaluate the absolute value expression, and finally add the results together.

Q3: What if the expression inside the square root is negative?

A3: If the expression inside the square root is negative, you will get an imaginary number as the result. For example, $\sqrt{-4}$ is equal to $2i$, where $i$ is the imaginary unit.

Q4: How do I handle expressions with multiple square roots and absolute values?

A4: When dealing with expressions that have multiple square roots and absolute values, you need to follow the order of operations (PEMDAS) carefully. Evaluate the expressions inside the square roots and absolute values one by one, and then add the results together.

Q5: Can I simplify expressions with square roots and absolute values?

A5: Yes, you can simplify expressions with square roots and absolute values by factoring out common terms or using algebraic identities. For example, $\sqrt{a^2} = |a|$.

Q6: How do I evaluate expressions with square roots and absolute values when the variables are negative?

A6: When the variables are negative, you need to be careful when evaluating the expressions. For example, $\sqrt{(-a)^2} = |a|$, but $\sqrt{a^2} = |a|$.

Q7: Can I use a calculator to evaluate expressions with square roots and absolute values?

A7: Yes, you can use a calculator to evaluate expressions with square roots and absolute values. However, make sure to check the calculator's settings to ensure that it is in the correct mode (e.g., decimal or fraction).

Q8: How do I check my work when evaluating expressions with square roots and absolute values?

A8: To check your work, plug in the values of the variables into the expression and evaluate it manually. This will help you ensure that your calculator or computer is giving you the correct answer.

Q9: Can I use algebraic identities to simplify expressions with square roots and absolute values?

A9: Yes, you can use algebraic identities to simplify expressions with square roots and absolute values. For example, $\sqrt{a^2} = |a|$.

Q10: How do I apply the order of operations (PEMDAS) when evaluating expressions with square roots and absolute values?

A10: To apply the order of operations (PEMDAS), follow these steps:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next (e.g., $\sqrt{a^2}$).
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.

Conclusion

Evaluating expressions with square roots and absolute values can be challenging, but with the right techniques and strategies, you can master them. Remember to follow the order of operations (PEMDAS), simplify expressions using algebraic identities, and check your work to ensure accuracy.

Key Takeaways

• Evaluate expressions inside the square root and absolute value first.
• Follow the order of operations (PEMDAS) carefully.
• Simplify expressions using algebraic identities.
• Check your work to ensure accuracy.

Related Problems

• Evaluate the expression $\sqrt{a^2-4} + |b|$ when $a=3$ and $b=-2$.
• Find the value of the expression $\sqrt{b^2-9} + |a|$ when $a=1$ and $b=4$.

Final Answer

The final answer is: $\boxed{5}$