Let's break down the expression step by step and solve it using the provided values for $a$ and $b$.

The given expression is:

$a^{-2} - b^2(b^0 - a) + |b|$

where:

• $a = \sqrt[3]{27}$
• $b + 5 = 3$, so $b = 3 - 5 = -2$

Step 1: Calculate the value of $a$

We are given that $a = \sqrt[3]{27}$, which is the cube root of 27. The cube root of 27 is:

$a = 3$

Step 2: Calculate the value of $b$

From the equation $b + 5 = 3$, we solve for $b$:

$b = -2$

Step 3: Substitute values into the expression

Substitute $a = 3$ and $b = -2$ into the given expression:

$3^{-2} - (-2)^2((-2)^0 - 3) + |-2|$

Step 4: Simplify the expression

• $3^{-2} = \frac{1}{3^2} = \frac{1}{9}$
• $(-2)^2 = 4$
• $(-2)^0 = 1$
• $1 - 3 = -2$
• The absolute value of $-2$ is $2$

Now, substitute these values into the expression:

$\frac{1}{9} - 4(-2) + 2$

Simplify further:

$\frac{1}{9} + 8 + 2$

$\frac{1}{9} + 10$

Finally, express the result as a fraction:

$10 = \frac{90}{9}$

$\frac{1}{9} + \frac{90}{9} = \frac{91}{9}$

Final Answer:

The value of the expression is $\frac{91}{9}$.

Would you like further details on the steps? Or do you have any other questions?

Here are some related questions:

1. How would the expression change if the powers of $a$ and $b$ were different?
2. What would happen if the absolute value were inside the parentheses?
3. How can you simplify similar expressions involving negative exponents?
4. How do you approach expressions with cube roots in general?
5. Can you calculate the cube root of other numbers using a similar method?

Tip: Always simplify expressions step by step and make sure to handle operations like exponents and absolute values first.