Forgetting to add the exponents

Oct 13, 2025 by windailys.com 108 views

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Introduction

When dealing with exponents, it's essential to understand the rules for simplifying expressions. In this article, we will focus on simplifying the expression $$x^5 \cdot x^7$$. This involves applying the rules of exponents, specifically the product of powers rule, to simplify the given expression.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, $$x^5$$ can be written as $$x \cdot x \cdot x \cdot x \cdot x$$. When we multiply two numbers with the same base, we can add their exponents. This is known as the product of powers rule.

The Product of Powers Rule

The product of powers rule states that when we multiply two numbers with the same base, we can add their exponents. Mathematically, this can be represented as:

a^m \cdot a^n = a^{m+n}

where $$a$$ is the base and $$m$$ and $$n$$ are the exponents.

Applying the Product of Powers Rule

Now that we understand the product of powers rule, let's apply it to the given expression $$x^5 \cdot x^7$$. Since both terms have the same base $$x$$, we can add their exponents.

x^5 \cdot x^7 = x^{5+7}

Simplifying the Expression

Using the product of powers rule, we can simplify the expression $$x^{5+7}$$ to $$x^{12}$$.

x^{5+7} = x^{12}

Conclusion

In this article, we simplified the expression $$x^5 \cdot x^7$$ using the product of powers rule. By adding the exponents, we were able to simplify the expression to $$x^{12}$$. This demonstrates the importance of understanding the rules of exponents in simplifying expressions.

Examples and Practice

To reinforce your understanding of the product of powers rule, try simplifying the following expressions:

$x^3 \cdot x^4$
$x^2 \cdot x^6$
$x^5 \cdot x^9$

Real-World Applications

The product of powers rule has numerous real-world applications in various fields, including:

Science: Exponents are used to represent the growth or decay of populations, chemical reactions, and physical phenomena.
Engineering: Exponents are used to represent the power consumption of electrical devices, the growth of populations, and the decay of materials.
Finance: Exponents are used to represent the growth of investments, the decay of assets, and the calculation of interest rates.

Common Mistakes

When simplifying expressions using the product of powers rule, it's essential to avoid common mistakes, such as:

Forgetting to add the exponents: Make sure to add the exponents when multiplying two numbers with the same base.
Using the wrong base: Ensure that both terms have the same base before applying the product of powers rule.
Not simplifying the expression: Simplify the expression by combining like terms and applying the product of powers rule.

Final Thoughts

In conclusion, simplifying the expression $$x^5 \cdot x^7$$ using the product of powers rule is a straightforward process. By adding the exponents, we were able to simplify the expression to $$x^{12}$$. This demonstrates the importance of understanding the rules of exponents in simplifying expressions. With practice and patience, you'll become proficient in simplifying expressions using the product of powers rule.

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Introduction

In our previous article, we simplified the expression $$x^5 \cdot x^7$$ using the product of powers rule. In this article, we will address some common questions and concerns related to simplifying expressions with exponents.

Q&A

Q: What is the product of powers rule?

A: The product of powers rule states that when we multiply two numbers with the same base, we can add their exponents. Mathematically, this can be represented as:

a^m \cdot a^n = a^{m+n}

where $$a$$ is the base and $$m$$ and $$n$$ are the exponents.

Q: How do I apply the product of powers rule?

A: To apply the product of powers rule, follow these steps:

Identify the base and exponents in both terms.
Check if both terms have the same base.
If they do, add the exponents.
Simplify the expression by combining like terms.

Q: What if the exponents are negative?

A: If the exponents are negative, you can still apply the product of powers rule. However, you need to be careful when adding negative numbers. For example:

x^{-5} \cdot x^{-7} = x^{-5-7} = x^{-12}

Q: Can I apply the product of powers rule to expressions with different bases?

A: No, the product of powers rule only applies to expressions with the same base. If the bases are different, you cannot add the exponents.

Q: How do I simplify expressions with multiple terms?

A: To simplify expressions with multiple terms, follow these steps:

Identify the base and exponents in each term.
Check if each term has the same base.
If they do, add the exponents.
Simplify the expression by combining like terms.

Q: What if I have an expression with a variable base?

A: If you have an expression with a variable base, you can still apply the product of powers rule. However, you need to be careful when adding the exponents. For example:

x^m \cdot x^n = x^{m+n}

Q: Can I apply the product of powers rule to expressions with fractions?

A: Yes, you can apply the product of powers rule to expressions with fractions. However, you need to be careful when adding the exponents. For example:

\frac{x^m}{x^n} = x^{m-n}

Q: How do I simplify expressions with exponents and fractions?

A: To simplify expressions with exponents and fractions, follow these steps:

Identify the base and exponents in each term.
Check if each term has the same base.
If they do, add the exponents.
Simplify the expression by combining like terms.