Problem 154 Simplify each expression. $$\l... [FREE SOLUTION]

Chapter 6: Problem 154

Simplify each expression. $$\left(-10 u^{2} v^{4}\right)^{3}$$

Short Answer

Expert verified

-1000u^6v^{12}

Step by step solution

Apply the Power Rule

Use the power rule $ (a^m)^n = a^{m \times n} $ to each part of the expression $ (-10u^2v^4)^3 $. This gives us $ (-10)^3 $, $ (u^2)^3 $, and $ (v^4)^3 $.

Calculate the Individual Powers

Compute the powers separately: $ (-10)^3 = -1000 $, $ (u^2)^3 = u^{2 \times 3} = u^6 $, and $ (v^4)^3 = v^{4 \times 3} = v^{12} $.

Combine the Results

Put everything together to get the simplified expression: $ -1000u^6v^{12} $.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power Rule in Algebra

One fundamental concept in algebra is the power rule. This simplifies expressions that involve exponents raised to another exponent. The power rule states that $(a^m)^n = a^{m \times n}$. This means we multiply the exponents to get a single exponent.
When applied to the expression $(-10u^2v^4)^3$, we need to raise each part of the expression by the power outside the parentheses. This includes the constant term \-10\, the exponent \u^2\, and the exponent \v^4\.
Using the power rule individually, we get $(-10)^3$, $(u^2)^3$, and $(v^4)^3$. By simplifying these, we proceed to the next steps of solving the expression.

Exponentiation

Exponentiation involves raising a number or variable to a power. When we have an expression like $(u^2)^3$, exponentiation helps us understand how to combine the exponents.
According to the power rule, we multiply the exponents: $(u^2)^3 = u^{2 \times 3} = u^6$. Similarly, $(v^4)^3 = v^{4 \times 3} = v^{12}$.
Don't forget to handle the constants separately. In our problem, $(-10)^3$ equals \-1000\.
Combining the results, we've simplified the exponents for each part of the expression, leading us to $-1000u^6v^{12}$.

Combining Like Terms

Combining like terms is another essential algebraic skill. In our simplified expression \-1000u^6v^{12}\, we don鈥檛 have more terms to combine this time since it鈥檚 all already simplified.
However, understanding this concept is crucial for more complex problems. Like terms are terms that have identical variable parts raised to the same powers. For example, $3x^2$ and $-5x^2$ are like terms, while $3x^2$ and $4x$ are not.
When combining like terms, we add or subtract their coefficients: $3x^2 - 5x^2 = -2x^2$. In your own work, always look for like terms and combine them to simplify your expressions further.

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91影视

Simplify each expression. $$\left(-10 u^{2} v^{4}\right)^{3}$$

Short Answer

Step by step solution

Apply the Power Rule

Calculate the Individual Powers

Combine the Results

Key Concepts

Power Rule in Algebra

Exponentiation

Combining Like Terms

One App. One Place for Learning.

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