Complete The Equation.${ (13) 2(13) {-4}(13)^5 = 13 }$
Understanding Exponents and Their Rules
Exponents are a fundamental concept in mathematics, used to represent repeated multiplication of a number. In the equation (13)2(13){-4}(13)^5 = 13, we are dealing with exponents and their properties. To simplify this equation, we need to understand the rules of exponents and how to apply them.
The Product of Powers Rule
One of the key rules of exponents is the product of powers rule, which states that when multiplying two or more numbers with the same base, we add their exponents. In other words, a^m * a^n = a^(m+n). This rule can be applied to the given equation to simplify the expression.
Applying the Product of Powers Rule
Let's apply the product of powers rule to the given equation:
(13)2(13){-4}(13)^5 = 13
Using the product of powers rule, we can rewrite the equation as:
(13)^(2 + (-4) + 5) = 13
Simplifying the Exponents
Now, let's simplify the exponents by adding them:
2 + (-4) + 5 = 3
So, the equation becomes:
(13)^3 = 13
Evaluating the Expression
Now that we have simplified the equation, let's evaluate the expression (13)^3. To do this, we need to multiply 13 by itself three times:
13 * 13 * 13 = 2197
Conclusion
In conclusion, the equation (13)2(13){-4}(13)^5 = 13 can be simplified using the product of powers rule. By applying this rule and simplifying the exponents, we arrive at the expression (13)^3 = 13. Evaluating this expression, we find that (13)^3 = 2197, which is indeed equal to 13.
The Importance of Understanding Exponents
Understanding exponents and their properties is crucial in mathematics, particularly in algebra and calculus. Exponents are used to represent repeated multiplication of a number, and their rules are essential in simplifying expressions and solving equations. By mastering the rules of exponents, we can solve complex equations and expressions with ease.
Real-World Applications of Exponents
Exponents have numerous real-world applications, including:
- Finance: Exponents are used to calculate compound interest and investment returns.
- Science: Exponents are used to represent the growth and decay of populations, chemical reactions, and physical phenomena.
- Engineering: Exponents are used to design and optimize systems, such as electrical circuits and mechanical systems.
Tips for Simplifying Exponents
Here are some tips for simplifying exponents:
- Use the product of powers rule: When multiplying two or more numbers with the same base, add their exponents.
- Use the power of a power rule: When raising a power to a power, multiply the exponents.
- Use the zero exponent rule: Any number raised to the zero power is equal to 1.
Common Mistakes to Avoid
Here are some common mistakes to avoid when simplifying exponents:
- Not using the product of powers rule: Failing to add exponents when multiplying numbers with the same base.
- Not using the power of a power rule: Failing to multiply exponents when raising a power to a power.
- Not using the zero exponent rule: Failing to recognize that any number raised to the zero power is equal to 1.
Conclusion
In conclusion, simplifying exponents is a crucial skill in mathematics, particularly in algebra and calculus. By mastering the rules of exponents and applying them correctly, we can solve complex equations and expressions with ease. Remember to use the product of powers rule, the power of a power rule, and the zero exponent rule to simplify exponents and avoid common mistakes.
Q: What is the product of powers rule?
A: The product of powers rule states that when multiplying two or more numbers with the same base, we add their exponents. In other words, a^m * a^n = a^(m+n).
Q: How do I apply the product of powers rule?
A: To apply the product of powers rule, simply add the exponents of the numbers with the same base. For example, (13)2(13){-4}(13)^5 = 13 can be simplified by adding the exponents: (13)^(2 + (-4) + 5) = 13.
Q: What is the power of a power rule?
A: The power of a power rule states that when raising a power to a power, we multiply the exponents. In other words, (am)n = a^(m*n).
Q: How do I apply the power of a power rule?
A: To apply the power of a power rule, simply multiply the exponents of the numbers with the same base. For example, (132)3 = 13^(2*3) = 13^6.
Q: What is the zero exponent rule?
A: The zero exponent rule states that any number raised to the zero power is equal to 1. In other words, a^0 = 1.
Q: How do I apply the zero exponent rule?
A: To apply the zero exponent rule, simply recognize that any number raised to the zero power is equal to 1. For example, 13^0 = 1.
Q: Can I simplify exponents with different bases?
A: No, the product of powers rule, the power of a power rule, and the zero exponent rule only apply to numbers with the same base. If you have numbers with different bases, you cannot simplify them using these rules.
Q: How do I simplify exponents with negative bases?
A: To simplify exponents with negative bases, you can use the rule that a^(-m) = 1/a^m. For example, (13)^(-4) = 1/(13^4).
Q: Can I simplify exponents with fractional bases?
A: No, the product of powers rule, the power of a power rule, and the zero exponent rule only apply to numbers with integer bases. If you have numbers with fractional bases, you cannot simplify them using these rules.
Q: How do I simplify exponents with complex bases?
A: To simplify exponents with complex bases, you can use the rule that a^(-m) = 1/a^m. For example, (13)^(-4) = 1/(13^4).
Q: Can I simplify exponents with multiple bases?
A: Yes, you can simplify exponents with multiple bases by using the product of powers rule. For example, (13)2(17)3 = 13^2 * 17^3.
Q: How do I simplify exponents with multiple exponents?
A: To simplify exponents with multiple exponents, you can use the product of powers rule. For example, (13)2(13)(-4)(13)^5 = 13^(2 + (-4) + 5).
Q: Can I simplify exponents with variables?
A: Yes, you can simplify exponents with variables by using the product of powers rule. For example, (x2)(x(-3)) = x^(2 + (-3)) = x^(-1).
Q: How do I simplify exponents with multiple variables?
A: To simplify exponents with multiple variables, you can use the product of powers rule. For example, (x2)(y3)(x^(-4)) = x^(2 + (-4)) * y^3 = x^(-2) * y^3.
Q: Can I simplify exponents with exponents in the denominator?
A: Yes, you can simplify exponents with exponents in the denominator by using the rule that a^(-m) = 1/a^m. For example, (13)^(-4) = 1/(13^4).
Q: How do I simplify exponents with fractions in the denominator?
A: To simplify exponents with fractions in the denominator, you can use the rule that a^(-m) = 1/a^m. For example, (13)^(-4/3) = 1/(13^(4/3)).
Q: Can I simplify exponents with complex fractions in the denominator?
A: Yes, you can simplify exponents with complex fractions in the denominator by using the rule that a^(-m) = 1/a^m. For example, (13)^(-4/3) = 1/(13^(4/3)).
Q: How do I simplify exponents with multiple fractions in the denominator?
A: To simplify exponents with multiple fractions in the denominator, you can use the rule that a^(-m) = 1/a^m. For example, (13)(-4/3)(17)(-2/3) = 1/(13^(4/3)) * 1/(17^(2/3)).
Q: Can I simplify exponents with variables in the denominator?
A: Yes, you can simplify exponents with variables in the denominator by using the rule that a^(-m) = 1/a^m. For example, (x^(-2)) = 1/(x^2).
Q: How do I simplify exponents with multiple variables in the denominator?
A: To simplify exponents with multiple variables in the denominator, you can use the rule that a^(-m) = 1/a^m. For example, (x(-2))(y(-3)) = 1/(x^2) * 1/(y^3).
Q: Can I simplify exponents with exponents in the numerator and denominator?
A: Yes, you can simplify exponents with exponents in the numerator and denominator by using the rule that a^(-m) = 1/a^m. For example, (132)/(134) = 1/(13^2).
Q: How do I simplify exponents with fractions in the numerator and denominator?
A: To simplify exponents with fractions in the numerator and denominator, you can use the rule that a^(-m) = 1/a^m. For example, (132)/(134) = 1/(13^2).
Q: Can I simplify exponents with complex fractions in the numerator and denominator?
A: Yes, you can simplify exponents with complex fractions in the numerator and denominator by using the rule that a^(-m) = 1/a^m. For example, (132)/(134) = 1/(13^2).
Q: How do I simplify exponents with multiple fractions in the numerator and denominator?
A: To simplify exponents with multiple fractions in the numerator and denominator, you can use the rule that a^(-m) = 1/a^m. For example, (132)/(134)(17^3) = 1/(13^2) * 1/(17^3).
Q: Can I simplify exponents with variables in the numerator and denominator?
A: Yes, you can simplify exponents with variables in the numerator and denominator by using the rule that a^(-m) = 1/a^m. For example, (x2)/(x4) = 1/(x^2).
Q: How do I simplify exponents with multiple variables in the numerator and denominator?
A: To simplify exponents with multiple variables in the numerator and denominator, you can use the rule that a^(-m) = 1/a^m. For example, (x2)/(x4)(y^3) = 1/(x^2) * 1/(y^3).
Q: Can I simplify exponents with exponents in the numerator and denominator that have different bases?
A: No, the product of powers rule, the power of a power rule, and the zero exponent rule only apply to numbers with the same base. If you have numbers with different bases, you cannot simplify them using these rules.
Q: How do I simplify exponents with fractions in the numerator and denominator that have different bases?
A: To simplify exponents with fractions in the numerator and denominator that have different bases, you can use the rule that a^(-m) = 1/a^m. For example, (132)/(174) = 1/(13^2).
Q: Can I simplify exponents with complex fractions in the numerator and denominator that have different bases?
A: Yes, you can simplify exponents with complex fractions in the numerator and denominator that have different bases by using the rule that a^(-m) = 1/a^m. For example, (132)/(174) = 1/(13^2).
Q: How do I simplify exponents with multiple fractions in the numerator and denominator that have different bases?
A: To simplify exponents with multiple fractions in the numerator and denominator that have different bases, you can use the rule that a^(-m) = 1/a^m. For example, (13^