Problem 80 For Exercises, simplify. $$\le... [FREE SOLUTION]

Chapter 9: Problem 80

For Exercises, simplify. $$\left(-2 x^{-5}\right) x^{7}$$

Short Answer

Expert verified

The simplified form of the expression $\left(-2 x^{-5}\right)x^{7}$ is $-2x^{2}$.

Step by step solution

Understand the Problem

We're asked to simplify $\left(-2 x^{-5}\right) x^{7}$. The exponent rule is needed here: when we multiply like bases, we add the exponents.

Apply the Exponent Rule

Apply the exponent rule by adding the exponents of $x$. Here, the exponents are -5 and 7. So, the new exponent of $x$ will be $ -5 + 7$.

Simplify the Expression

Perform the addition of the exponents, which gives us $ -5 + 7 = 2$. Hence, the expression simplifies to $-2x^{2}$.

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Key Concepts

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

Exponent Rules

Exponent rules are fundamental in simplifying expressions involving powers. They help us manage operations with multiple exponents efficiently. The key rule for this exercise is: when multiplying exponents with the same base, we add their powers.
This is known as the *Product of Powers Rule*. If you have terms like $a^m \cdot a^n$, you simply add the exponents together, resulting in $a^{m+n}$.
Make sure to recognize this rule as it is often used in algebraic manipulation, especially when dealing with polynomial or algebraic expressions.

Like Bases

"Like bases" refers to terms that have the same base number or variable. In our exercise, $-2x^{-5}$ and $x^7$ both have the base $x$.
Identifying like bases is crucial for simplifying expressions, as it allows you to apply exponent rules effectively.
When you recognize like bases, you can go ahead and combine them by simply adding the exponents, **only if they are being multiplied together**.

Multiplication of Exponents

When multiplying expressions with exponents, such as $a^m\times a^n$, it's essential to apply the multiplication rules for exponents.
This involves adding the exponents only if the bases are the same.

If you have $x^{-5}$ and $x^{7}$, simply add $-5$ and $7$ to get the new exponent.
In our case, $-5 + 7 = 2$, resulting in $x^2$.

This reduces complex expressions to simpler forms, making calculations more straightforward.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators (like addition or multiplication). Simplifying these expressions often requires combining like terms and applying various rules like the exponent rules.
In the given exercise $(-2x^{-5}) x^7$, the goal is to express it in its simplest form.

By applying the product of powers rule and focusing on like bases, you can streamline the expression.
This simplification transforms the expression into $-2x^2$, a much simpler algebraic form.

With practice, simplifying algebraic expressions becomes an automatic process, crucial for solving more complex math problems.

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91影视

For Exercises, simplify. $$\left(-2 x^{-5}\right) x^{7}$$

Short Answer

Step by step solution

Understand the Problem

Apply the Exponent Rule

Simplify the Expression

Key Concepts

Exponent Rules

Like Bases

Multiplication of Exponents

Algebraic Expressions

One App. One Place for Learning.

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