Q. Evaluate \(5\lvert x^3 – 2\rvert + 7\) when \(x = -2\).

Q: What is 5|x^3 - 2| + 7 when x = -2?

Compute (-2)^3 = -8; inside: -8-2=-10; |-10|=10; 5\cdot10=50; 50+7=57. The value is 57.

Q: Why is (-2)^3 = -8 and not 8?

Odd powers preserve sign, so (-2)^3 = -2\cdot-2\cdot-2 = -8. Parentheses matter: -2^3 without parentheses equals -(2^3) = -8 by standard order, but write (-2)^3 to be clear.

Q: How do I evaluate expressions with absolute value like |x^3-2|?

First compute x^3-2. Then take absolute value: |a|=a if a \ge 0, and |a|=-a if a < 0. Finally apply remaining operations (multiply, add).

Q: Could I simplify 5|x^3-2|+7 algebraically before substituting?

Not meaningfully: absolute value blocks algebraic cancellation. You can state piecewise: 5(x^3-2)+7 for x^3 \ge 2 and 5(2-x^3)+7 for x^3 < 2, then substitute.

Q: Is the domain of 5|x^3-2|+7 all real numbers?

Yes. x^3-2 and absolute value are defined for every real x, so the expression is defined for all real numbers.

Q: What's the difference between |x^3|-2 and |x^3-2|?

What's the difference between |x^3|-2 and |x^3-2|?

Q: How can I avoid sign mistakes with negative bases and exponents?

Always use parentheses for negative bases, e.g., (-2)^3. Remember exponentiation happens before unary minus only if parentheses are absent: -2^3 = -(2^3). When in doubt, compute stepwise: evaluate power first with sign clarified.

Answer

Compute \(5\lvert x^{3}-2\rvert+7\) for \(x=-2\):

\(x^{3}=(-2)^{3}=-8\)

\(x^{3}-2=-8-2=-10\)

\(\lvert x^{3}-2\rvert=10\)

\(5\lvert x^{3}-2\rvert+7=5\cdot 10+7=57\)

Final result: 57.

Detailed Explanation

Write the expression to evaluate:

\(5\lvert x^{3} – 2\rvert + 7\)
Substitute \(x = -2\) into the expression. When substituting, place the value in parentheses because the exponent applies to the whole number \(-2\):

\(5\lvert(-2)^{3} – 2\rvert + 7\)
Compute the cube \((-2)^{3}\). Raising \(-2\) to the third power gives:

\((-2)^{3} = -8\)

So the expression becomes:

\(5\lvert -8 – 2\rvert + 7\)
Evaluate the arithmetic inside the absolute value:

\(-8 – 2 = -10\)

Now we have:

\(5\lvert -10\rvert + 7\)
Apply the absolute value. The absolute value of a negative number is its positive counterpart:

\(\lvert -10\rvert = 10\)

So the expression becomes:

\(5 \times 10 + 7\)
Multiply and then add:

\(5 \times 10 = 50\)

\(50 + 7 = 57\)
Final answer:

\(57\)

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FAQs

What is \(5|x^3 - 2| + 7\) when \(x = -2\)?

Compute \( (-2)^3 = -8\); inside: \(-8-2=-10\); \(|-10|=10\); \(5\cdot10=50\); \(50+7=57\). The value is 57.

Why is \((-2)^3 = -8\) and not 8?

Odd powers preserve sign, so \((-2)^3 = -2\cdot-2\cdot-2 = -8\). Parentheses matter: \(-2^3\) without parentheses equals \(-(2^3) = -8\) by standard order, but write \((-2)^3\) to be clear.

How do I evaluate expressions with absolute value like \(|x^3-2|\)?

First compute \(x^3-2\). Then take absolute value: \(|a|=a\) if \(a \ge 0\), and \(|a|=-a\) if \(a < 0\). Finally apply remaining operations (multiply, add).

Could I simplify \(5|x^3-2|+7\) algebraically before substituting?

Not meaningfully: absolute value blocks algebraic cancellation. You can state piecewise: \(5(x^3-2)+7\) for \(x^3 \ge 2\) and \(5(2-x^3)+7\) for \(x^3 < 2\), then substitute.

Is the domain of \(5|x^3-2|+7\) all real numbers?

Yes. \(x^3-2\) and absolute value are defined for every real \(x\), so the expression is defined for all real numbers.

What's the difference between \(|x^3|-2\) and \(|x^3-2|\)?

How can I avoid sign mistakes with negative bases and exponents?

Always use parentheses for negative bases, e.g., \((-2)^3\). Remember exponentiation happens before unary minus only if parentheses are absent: \(-2^3 = -(2^3)\). When in doubt, compute stepwise: evaluate power first with sign clarified.

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