Problem 89 Simplify each expression. $$ ... [FREE SOLUTION]

Chapter 0: Problem 89

Simplify each expression. $$ 2 y^{2}-\left[13-\frac{2}{3}\left(6 y^{2}-9\right)-10\right]+9 $$

Short Answer

Expert verified

6y^2

Step by step solution

- Distribute the Fraction Inside the Parentheses

First, handle the fraction inside the parentheses $6y^2 - 9$. Multiply each term by \frac{2}{3}\: \[ \frac{2}{3} \times 6y^2 - \frac{2}{3} \times 9 = 4y^2 - 6 \]

- Simplify Inside the Brackets

Next, simplify the expression inside the brackets by performing the subtraction \[ 13 - (4y^2 - 6) - 10 \]. Distribute the negative sign: \[ 13 - 4y^2 + 6 - 10 \]. Combine like terms: \[ 13 + 6 - 10 - 4y^2 = 9 - 4y^2 \]

- Combine with the Rest of the Expression

Now, simplify the full expression by combining with other terms \[ 2y^2 - (9 - 4y^2) + 9 \]. Distribute the negative sign again: \[ 2y^2 - 9 + 4y^2 + 9 \]. Combine like terms: \[ 2y^2 + 4y^2 - 9 + 9 = 6y^2 \]

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Key Concepts

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

Distributing Fractions

When you encounter a fraction outside of parentheses, you need to distribute it to every term inside the parentheses. For instance, in the expression $\frac{2}{3}(6y^2 - 9)$, distribute $ \frac{2}{3} $ to both 6y\raisebox{0.25em}{$^{2}$} and -9.

Multiply as follows:

$ \frac{2}{3} \times 6y^2 = 4y^2 $
$ \frac{2}{3} \times -9 = -6 $

So, $ \frac{2}{3}(6y^2 - 9) = 4y^2 - 6 $.

Remember, every time you distribute a fraction or any term over a subtraction or addition within parentheses, each term inside gets multiplied by the fraction.

Combining Like Terms

After distributing, the next important step is to combine like terms.
This means you group terms that have the same variable raised to the same power.
For instance, in the expression $ 13 - 4y^2 + 6 - 10 $, you combine constant terms and variable terms separately.

First, combine constants: 13 + 6 - 10, which results in 9.
Then include the variable term: $ -4y^2 $.

This gets you $ 9 - 4y^2 $.
Breaking down complex expressions by combining like terms makes the algebra much simpler.

Algebraic Simplification

The final step is to simplify the entire algebraic expression.
Simplification often involves both distributing and combining like terms.
For the given problem, the expression to simplify is $ 2y^2 - (9 - 4y^2) + 9 $.

First, distribute the negative sign: $ 2y^2 - 9 + 4y^2 + 9 $. Then combine the like terms:

Combine variable terms $ 2y^2 + 4y^2 = 6y^2 $.
Combine constant terms -9 + 9 = 0.

This simplifies the expression to $ 6y^2 $.
Every step in the simplification process makes the expression cleaner and easier to understand.
Remember these steps to tackle any algebraic expression with confidence.

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91影视

Simplify each expression. $$ 2 y^{2}-\left[13-\frac{2}{3}\left(6 y^{2}-9\right)-10\right]+9 $$

Short Answer

Step by step solution

- Distribute the Fraction Inside the Parentheses

- Simplify Inside the Brackets

- Combine with the Rest of the Expression

Key Concepts

Distributing Fractions

Combining Like Terms

Algebraic Simplification

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