Problem 81 Simplify. $$3 x^{2}+(3 x)^{2}$... [FREE SOLUTION]

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Chapter 9: Problem 81

Simplify. $$3 x^{2}+(3 x)^{2}$$

Short Answer

Expert verified

The simplified expression is $12x^2$.

Step by step solution

Identify Terms with Exponents

In this problem, there are two terms with exponents: $3x^2$ and $(3x)^2$.

Simplify Each Term

The first term $3x^2$ doesn't need any further simplification. For the second term, $(3x)^2$, we must square both the 3 and the $x$, which gives $9x^2$

Add Like Terms

Finally, we can add the like terms: $3x^2$ and $9x^2$, which gives $12x^2$ as the simplified expression.

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Key Concepts

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

Exponents

Exponents are an essential concept in algebra that simplify the process of multiplication. They are written as a small number to the upper right of a base number or variable. When you see something like $x^n$, it means you multiply $x$ by itself $n$ times. In the problem $(3x)^2$, the exponent 2 applies to everything in the parentheses. This means both 3 and $x$ must be squared. So it becomes $3^2$, which is 9, and $x^2$. Thus, $(3x)^2 = 9x^2$.

Exponents make calculations easier by indicating how many times to multiply a number by itself.
In expressions, pay attention to where the exponent is placed; it applies only to the base directly beneath it.

Keep in mind that simplifying an expression with exponents can involve expanding and then combining like terms, which we'll discuss next.

Like Terms

Like terms in algebra are terms that have the same variable part and the same exponent. Only like terms can be combined through addition and subtraction. In the expression $3x^2 + 9x^2$, the terms $3x^2$ and $9x^2$ are like terms because they both have $x^2$ as their variable part. This allows them to be added together; the coefficients (3 and 9) are simply added to get 12, resulting in $12x^2$.

Like terms must have identical variables and exponents.
You combine like terms by adding or subtracting their coefficients.

Combining like terms makes expressions simpler and easier to read.

Algebraic Expressions

An algebraic expression is a combination of variables, numbers, and operations (like addition and multiplication). Simplifying algebraic expressions often involves dealing with exponents and combining like terms.
In our original problem, the expression $3x^2 + (3x)^2$ includes different elements:

$3x^2$, which is a term with a constant (3) and a variable $x$ raised to an exponent (2).
$(3x)^2$, which requires expansion by squaring both the 3 and the $x$.

Algebraic expressions can be simplified by carefully applying rules for exponents and combining like terms, as demonstrated in the solution, which results in $12x^2$. Understanding how to manipulate and simplify these expressions is key in algebra.

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91影视

Simplify. $$3 x^{2}+(3 x)^{2}$$

Short Answer

Step by step solution

Identify Terms with Exponents

Simplify Each Term

Add Like Terms

Key Concepts

Exponents

Like Terms

Algebraic Expressions

One App. One Place for Learning.

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