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Questions and Answers
According to the product rule of exponents, what is the result of multiplying 4^3 by 4^5?
According to the product rule of exponents, what is the result of multiplying 4^3 by 4^5?
If you have to divide 5^7 by 5^4, according to the quotient rule, what is the simplified form?
If you have to divide 5^7 by 5^4, according to the quotient rule, what is the simplified form?
What is the result of (6^2)^3 according to the power of a power rule?
What is the result of (6^2)^3 according to the power of a power rule?
If you need to multiply 2^6 by 2^3, what is the result following the product rule?
If you need to multiply 2^6 by 2^3, what is the result following the product rule?
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If you have to divide 7^5 by 7^8, what would be the outcome using the quotient rule?
If you have to divide 7^5 by 7^8, what would be the outcome using the quotient rule?
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Study Notes
Laws of Exponents
Exponents and powers refer to mathematical operations where you raise a number to some power. In this context, we look into three fundamental rules known as the laws of exponents. These rules can help us simplify expressions involving exponents by performing arithmetic on the bases and exponents separately. Let's dive deeper into these rules:
Product Rule
The product rule states that if two numbers have the same base, you can multiply their exponents and divide the result by the exponent of the original base. Mathematically, it is expressed as follows:
(a^m) * (a^n) = a^(m+n)
For example, let's take a^(m) = 2^4 and a^(n) = 2^3. According to the product rule, their sum would be a^(m + n), which equals 2^7.
Quotient Rule - Division
The quotient rule states that when dividing expressions with different bases, you subtract one exponent from another and change the new base to the original basis. Mathematically, it is expressed as:
(a^m) / (a^n) = a^(m-n)
For instance, consider 3^4 / 3^3. According to the quotient rule, we subtract the exponent of the denominator from that of the numerator, resulting in 3^1.
Power of a Power Rule
The power of a power rule states that you can raise an exponent to another power by multiplying the exponents. Mathematically, it is expressed as:
(a^m)^n = a^(mn)
For example, let's consider the expression (2^3)^2. According to the power of a power rule, we multiply the exponents, resulting in 2^(3*2) or 2^6.
Zero Exponent Rule
The zero exponent rule states that any number raised to the power of 0 is equal to 1. Mathematically, it is expressed as:
a^0 = 1
For instance, 5^0 equals 1.
Negative Exponent Rule
The negative exponent rule states that if the exponent is a negative integer, you multiply the base by the reciprocal of the absolute value of the exponent. Mathematically, it is expressed as:
a^(-m) = 1/a^m
For example, let's consider 3^(-2). According to the negative exponent rule, we take the reciprocal of 2, which is 1/2, and multiply it by 3. This results in the value 3^-2 = 3/2.
These laws of exponents help simplify complex expressions involving exponents, making calculations more efficient and manageable.
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Description
Explore and test your knowledge on the laws of exponents, including the product rule, quotient rule, power of a power rule, zero exponent rule, and negative exponent rule. Understand how these rules simplify expressions with exponents and enhance your arithmetic skills.
