Problem 53 - Intermediate Algebra

Question

Simplify each expression. Write answers using positive exponents. $$ \left(-x^{2}\right)^{5} y^{7} y^{3} x^{-2} y^{0} $$

Step-by-Step Solution

Answer

The simplified expression is \\(-x^8 y^{10}\\).

Step 1: Apply the power rule

First, apply the power rule for exponents to \(-x^2\)^5. According to the power rule, \((a^m)^n = a^{m\times n}\). So \((-x^2)^5 = (-1)^5 \times (x^2)^5\). As \((-1)^5 = -1\), this gives \(-x^{10}\).

Step 2: Combine like terms

Now, combine all the terms involving the same base. Start with the \(y\) terms: \(y^7 \times y^3\). According to the multiplication rule, \(a^m \times a^n = a^{m+n}\). So, \(y^7 \times y^3 = y^{7+3} = y^{10}\).

Step 3: Simplify the expression

Combine the newly found expressions and simplify. The expression simplifies to \(-x^{10} \times x^{-2} \times y^{10}\). For \(x\) terms, use the multiplication rule again: \(-x^{10} \times x^{-2} = -x^{10 + (-2)} = -x^{8}\).

Step 4: Eliminate zero exponent

Note that any term with a zero exponent simplifies to 1. Since \(y^0 = 1\), it does not change the expression when multiplied. Thus, it can be ignored in the final simplified expression.

Step 5: Write the final expression

Substitute the simplified parts back into the expression. The final simplified expression is \(-x^8 y^{10}\). Ensure that exponents are positive and the expression is written neatly.

Key Concepts

Exponent Rules

Understanding the Exponent Rules is crucial for simplifying expressions involving powers. When dealing with exponents, it helps to know the power rule, which states that

• \((a^m)^n = a^{m \times n}\).

This means that when you raise a power to another power, you multiply the exponents.
For example, applying this to \((-x^2)^5\), using

• \((-x^2)^5 = (-1)^5 \times (x^2)^5\),
• \((x^2)^5 = x^{2 \times 5} = x^{10}\),
• and \((-1)^5 = -1\), so we have \(-x^{10}\).

Once the power rule is applied, it helps to proceed with simplifying the expression by combining like terms.

Simplifying Expressions

When you simplify expressions, you mainly aim to condense them into the simplest form, often by combining like terms and reducing powers. Combine terms with the same base by using the following rule:

• \(a^m \times a^n = a^{m+n}\).

Let's take the initial expression:

• x terms: \(-x^{10} \times x^{-2} = -x^{10 + (-2)} = -x^{8}\)
• y terms: \(y^7 \times y^3 = y^{7+3} = y^{10}\).

It's important to identify terms with zero powers since any number raised to the zero power is 1, and multiplying by 1 doesn't change the expression's value. Hence, terms like \(y^0 = 1\) can be ignored. The expression simplifies to just \(-x^8 y^{10}\) without needing further changes.

Positive Exponents

While working to ensure all exponents are positive, it's essential to remember that negative exponents indicate the reciprocal of the base raised to the corresponding positive exponent.

• \(a^{-m} = \frac{1}{a^m}\).

However, in expressions already simplified to positive exponents, there's typically no need for additional steps.
In our example,

• in transforming \(-x^8 y^{10}\), all exponents are inherently positive,
• thus enabling a neatly written final form where every exponent remains positive and manageable.

Closing gaps and ensuring expressions comply with these rules can facilitate a smoother math experience.