Simplify The Expression:$\[ \frac{a^6 \times A^2}{a^{10}} \\]

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Introduction

When dealing with exponents, it's essential to understand the rules of exponentiation to simplify complex expressions. In this article, we will focus on simplifying the given expression a6×a2a10\frac{a^6 \times a^2}{a^{10}} using the properties of exponents.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, a3a^3 can be written as a×a×aa \times a \times a. When we multiply two numbers with the same base, we add their exponents. For instance, a3×a2=a3+2=a5a^3 \times a^2 = a^{3+2} = a^5.

Simplifying the Expression

To simplify the given expression, we need to apply the rules of exponentiation. The expression is a6×a2a10\frac{a^6 \times a^2}{a^{10}}. We can start by combining the exponents in the numerator using the rule am×an=am+na^m \times a^n = a^{m+n}.

a^6 × a^2 = a^(6+2) = a^8

Applying the Quotient Rule

Now that we have simplified the numerator, we can apply the quotient rule to simplify the expression. The quotient rule states that when we divide two numbers with the same base, we subtract their exponents. In this case, we have a8a10\frac{a^8}{a^{10}}.

a^8 ÷ a^10 = a^(8-10) = a^(-2)

Understanding Negative Exponents

A negative exponent is a shorthand way of representing a fraction. For example, a−2a^{-2} can be written as 1a2\frac{1}{a^2}. When we have a negative exponent, we can rewrite it as a fraction by flipping the base.

Simplifying the Expression Further

Now that we have simplified the expression to a−2a^{-2}, we can rewrite it as a fraction by flipping the base.

a^(-2) = 1/a^2

Conclusion

In this article, we simplified the expression a6×a2a10\frac{a^6 \times a^2}{a^{10}} using the properties of exponents. We applied the rules of exponentiation, including the product rule and the quotient rule, to simplify the expression. We also understood the concept of negative exponents and how to rewrite them as fractions.

Final Answer

The final answer to the expression a6×a2a10\frac{a^6 \times a^2}{a^{10}} is 1a2\frac{1}{a^2}.

Frequently Asked Questions

Q: What is the rule for multiplying exponents with the same base?

A: When we multiply two numbers with the same base, we add their exponents. For example, a3×a2=a3+2=a5a^3 \times a^2 = a^{3+2} = a^5.

Q: What is the rule for dividing exponents with the same base?

A: When we divide two numbers with the same base, we subtract their exponents. For example, a8÷a10=a8−10=a−2a^8 \div a^{10} = a^{8-10} = a^{-2}.

Q: What is a negative exponent?

A: A negative exponent is a shorthand way of representing a fraction. For example, a−2a^{-2} can be written as 1a2\frac{1}{a^2}.

Additional Resources

Q&A: Simplifying Exponents

Q: What is the rule for multiplying exponents with the same base?

A: When we multiply two numbers with the same base, we add their exponents. For example, a3×a2=a3+2=a5a^3 \times a^2 = a^{3+2} = a^5.

Q: What is the rule for dividing exponents with the same base?

A: When we divide two numbers with the same base, we subtract their exponents. For example, a8÷a10=a8−10=a−2a^8 \div a^{10} = a^{8-10} = a^{-2}.

Q: What is a negative exponent?

A: A negative exponent is a shorthand way of representing a fraction. For example, a−2a^{-2} can be written as 1a2\frac{1}{a^2}.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you need to apply the rules of exponentiation. First, combine the exponents in the numerator using the rule am×an=am+na^m \times a^n = a^{m+n}. Then, apply the quotient rule to simplify the expression.

Q: What is the quotient rule?

A: The quotient rule states that when we divide two numbers with the same base, we subtract their exponents. For example, a8÷a10=a8−10=a−2a^8 \div a^{10} = a^{8-10} = a^{-2}.

Q: How do I rewrite a negative exponent as a fraction?

A: To rewrite a negative exponent as a fraction, you need to flip the base. For example, a−2a^{-2} can be written as 1a2\frac{1}{a^2}.

Q: What is the final answer to the expression a6×a2a10\frac{a^6 \times a^2}{a^{10}}?

A: The final answer to the expression a6×a2a10\frac{a^6 \times a^2}{a^{10}} is 1a2\frac{1}{a^2}.

Q: Can you provide more examples of simplifying expressions with exponents?

A: Here are a few more examples:

  • a3×a4a2=a3+4−2=a5\frac{a^3 \times a^4}{a^2} = a^{3+4-2} = a^5
  • a2÷a3a4=a2−3−4=a−5\frac{a^2 \div a^3}{a^4} = a^{2-3-4} = a^{-5}
  • a5×a2a3=a5+2−3=a4\frac{a^5 \times a^2}{a^3} = a^{5+2-3} = a^4

Q: Where can I learn more about exponents and simplifying expressions?

A: There are many online resources available to learn more about exponents and simplifying expressions. Some popular resources include: