Given That \[$a, B\$\], And \[$c\$\] Are Positive Integers Such That \[$a^b = X\$\] And \[$c^b = Y\$\], Then What Is The Value Of \[$x \cdot Y\$\]?A) B) \[$(ac)^b\$\]C) D) \[$(ac)^{b^2}\$\]E)
Introduction
In mathematics, exponents play a crucial role in representing large numbers in a compact form. Given that {a, b$}$, and {c$}$ are positive integers such that {a^b = x$}$ and {c^b = y$}$, we are tasked with finding the value of {x \cdot y$}$. This problem requires a deep understanding of exponent properties and their applications.
Understanding Exponents
Exponents are a shorthand way of representing repeated multiplication. For example, {a^b$}$ represents the product of {a$}$ multiplied by itself {b$}$ times. In other words, {a^b = a \cdot a \cdot a \cdot ... \cdot a$}$ ({b$}$ times). This notation allows us to express large numbers in a more manageable form.
The Relationship Between Exponents and Multiplication
When we multiply two numbers with the same exponent, we can combine their bases and keep the exponent the same. This property is known as the product of powers rule. Mathematically, this can be represented as:
{(a^b) \cdot (c^b) = (a \cdot c)^b$}$
This rule allows us to simplify expressions involving exponents and multiplication.
Applying the Product of Powers Rule
In our problem, we are given that {a^b = x$}$ and {c^b = y$}$. We want to find the value of {x \cdot y$}$. Using the product of powers rule, we can rewrite the expression as:
{x \cdot y = (a^b) \cdot (c^b) = (a \cdot c)^b$}$
This simplifies the expression and allows us to find the value of {x \cdot y$}$.
Conclusion
In conclusion, the value of {x \cdot y$}$ is {(ac)^b$}$. This result is obtained by applying the product of powers rule, which states that when we multiply two numbers with the same exponent, we can combine their bases and keep the exponent the same.
The Final Answer
The final answer is (ac)^b.
Additional Examples
To further illustrate the concept, let's consider a few additional examples:
- If ${2^3 = 8\$} and ${3^3 = 27\$}, then what is the value of ${8 \cdot 27\$}?
- If ${4^2 = 16\$} and ${5^2 = 25\$}, then what is the value of ${16 \cdot 25\$}?
In both cases, we can apply the product of powers rule to simplify the expression and find the value of the product.
Common Mistakes to Avoid
When working with exponents and multiplication, it's essential to remember the following common mistakes:
- Not applying the product of powers rule when multiplying numbers with the same exponent.
- Not simplifying expressions involving exponents and multiplication.
By avoiding these common mistakes, you can ensure that your calculations are accurate and reliable.
Real-World Applications
Exponents and multiplication have numerous real-world applications in various fields, including:
- Finance: Exponents are used to calculate compound interest and investment returns.
- Science: Exponents are used to represent large numbers in scientific notation.
- Engineering: Exponents are used to calculate stress and strain in materials.
Q&A: Exponents and Multiplication
Q: What is the product of powers rule?
A: The product of powers rule states that when we multiply two numbers with the same exponent, we can combine their bases and keep the exponent the same. Mathematically, this can be represented as:
{(a^b) \cdot (c^b) = (a \cdot c)^b$}$
Q: How do I apply the product of powers rule?
A: To apply the product of powers rule, simply multiply the bases of the two numbers and keep the exponent the same. For example, if we have ${2^3\$} and ${3^3\$}, we can apply the product of powers rule as follows:
{(2^3) \cdot (3^3) = (2 \cdot 3)^3 = 6^3$}$
Q: What is the difference between {(a^b)$] and [$(a \cdot b)$]?
A: [a$}$ multiplied by itself {b$}$ times, while {(a \cdot b)$] represents the product of [a\$} and {b$}$. For example, if {a = 2$}$ and {b = 3$}$, then:
{(2^3) = 2 \cdot 2 \cdot 2 = 8$}{(2 \cdot 3) = 6\$}
Q: Can I apply the product of powers rule to numbers with different exponents?
A: No, the product of powers rule only applies to numbers with the same exponent. If the exponents are different, you cannot combine the bases and keep the exponent the same.
Q: How do I simplify expressions involving exponents and multiplication?
A: To simplify expressions involving exponents and multiplication, apply the product of powers rule and combine the bases of the two numbers. For example, if we have {(2^3) \cdot (3^2)$}$, we can simplify the expression as follows:
{(2^3) \cdot (3^2) = (2 \cdot 3)^{3+2} = 6^5$}$
Q: What are some common mistakes to avoid when working with exponents and multiplication?
A: Some common mistakes to avoid when working with exponents and multiplication include:
- Not applying the product of powers rule when multiplying numbers with the same exponent.
- Not simplifying expressions involving exponents and multiplication.
- Confusing the product of powers rule with the quotient of powers rule.
Q: How do I apply the quotient of powers rule?
A: The quotient of powers rule states that when we divide two numbers with the same exponent, we can divide their bases and keep the exponent the same. Mathematically, this can be represented as:
{(a^b) \div (c^b) = (a \div c)^b$}$
For example, if we have {(2^3) \div (3^3)$}$, we can apply the quotient of powers rule as follows:
{(2^3) \div (3^3) = (2 \div 3)^3 = (2/3)^3$}$
Q: What are some real-world applications of exponents and multiplication?
A: Exponents and multiplication have numerous real-world applications in various fields, including:
- Finance: Exponents are used to calculate compound interest and investment returns.
- Science: Exponents are used to represent large numbers in scientific notation.
- Engineering: Exponents are used to calculate stress and strain in materials.
In conclusion, the product of powers rule is a fundamental concept in mathematics that allows us to simplify expressions involving exponents and multiplication. By understanding and applying this rule, we can solve a wide range of problems and make sense of complex mathematical concepts.