Solving For K In Product Of Powers Equation

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In the realm of mathematics, exponents and powers play a crucial role in expressing and manipulating numbers. When dealing with the product of powers, understanding the rules of exponents is essential for simplifying expressions and solving equations. This article delves into the intricacies of finding the value of 'k' in the product of powers, providing a step-by-step approach to solve the equation:

$10^{-3} imes 10 imes 10^k = 10^{-3}$.

We will explore the fundamental concepts of exponents, demonstrate the application of exponent rules, and guide you through the process of isolating 'k' to determine its value. This comprehensive guide will equip you with the knowledge and skills to confidently tackle similar problems involving exponents and powers.

Understanding Exponents and Powers

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Before we dive into solving the equation, let's lay the foundation by understanding the concept of exponents and powers. An exponent indicates the number of times a base number is multiplied by itself. For instance, in the expression $10^3$, 10 is the base, and 3 is the exponent. This means 10 is multiplied by itself three times: $10 imes 10 imes 10 = 1000$. The result, 1000, is the power.

Exponents can be positive, negative, or zero. A positive exponent indicates repeated multiplication, while a negative exponent indicates the reciprocal of the base raised to the positive exponent. For example, $10^{-3}$ is equivalent to $\frac{1}{10^3}$, which equals $\frac{1}{1000}$ or 0.001. A zero exponent means the base raised to the power of zero is equal to 1. For example, $10^0 = 1$.

Understanding these fundamental concepts is crucial for manipulating expressions involving exponents and powers. We can now delve into the rules of exponents, which will help us simplify and solve the given equation.

Rules of Exponents

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When dealing with the product of powers, several rules govern how exponents interact. These rules are essential for simplifying expressions and solving equations. Let's explore the key rules that are relevant to our problem:

1. Product of Powers Rule

The product of powers rule states that when multiplying powers with the same base, you add the exponents. Mathematically, this can be expressed as: $a^m imes a^n = a^{m+n}$, where 'a' is the base, and 'm' and 'n' are the exponents.

For example, $10^2 imes 10^3 = 10^{2+3} = 10^5$. This rule simplifies the multiplication of powers with the same base by combining the exponents.

2. Quotient of Powers Rule

The quotient of powers rule states that when dividing powers with the same base, you subtract the exponents. Mathematically, this can be expressed as: $\frac{a^m}{a^n} = a^{m-n}$, where 'a' is the base, and 'm' and 'n' are the exponents.

For example, $\frac{10^5}{10^2} = 10^{5-2} = 10^3$. This rule simplifies the division of powers with the same base by subtracting the exponents.

3. Power of a Power Rule

The power of a power rule states that when raising a power to another power, you multiply the exponents. Mathematically, this can be expressed as: $(a^m)^n = a^{m imes n}$, where 'a' is the base, and 'm' and 'n' are the exponents.

For example, $(10^2)^3 = 10^{2 imes 3} = 10^6$. This rule simplifies raising a power to another power by multiplying the exponents.

4. Power of a Product Rule

The power of a product rule states that when raising a product to a power, you raise each factor in the product to that power. Mathematically, this can be expressed as: $(ab)^n = a^n b^n$, where 'a' and 'b' are the factors, and 'n' is the exponent.

For example, $(2 imes 5)^3 = 2^3 imes 5^3 = 8 imes 125 = 1000$. This rule simplifies raising a product to a power by distributing the exponent to each factor.

5. Power of a Quotient Rule

The power of a quotient rule states that when raising a quotient to a power, you raise both the numerator and the denominator to that power. Mathematically, this can be expressed as: $(\frac{a}{b})^n = \frac{a^n}{b^n}$, where 'a' is the numerator, 'b' is the denominator, and 'n' is the exponent.

For example, $(\frac{10}{2})^3 = \frac{10^3}{2^3} = \frac{1000}{8} = 125$. This rule simplifies raising a quotient to a power by distributing the exponent to both the numerator and the denominator.

These rules of exponents provide a powerful toolkit for simplifying expressions and solving equations involving powers. In the next section, we will apply these rules to solve the given equation and find the value of 'k'.

Solving for k in the Product of Powers

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Now, let's apply our understanding of exponents and their rules to solve the equation:

$10^{-3} imes 10 imes 10^k = 10^{-3}$

Our goal is to isolate 'k' and determine its value. We can achieve this by systematically applying the rules of exponents.

Step 1: Simplify the Left Side of the Equation

The left side of the equation involves the product of powers with the same base (10). We can use the product of powers rule to simplify this expression. Recall that the product of powers rule states that when multiplying powers with the same base, you add the exponents. Therefore, we have:

$10^{-3} imes 10 imes 10^k = 10^{-3 + 1 + k} = 10^{k - 2}$

Here, we added the exponents -3, 1 (since $10 = 10^1$), and 'k'.

Step 2: Rewrite the Equation

Now, we can rewrite the original equation with the simplified left side:

$10^{k - 2} = 10^{-3}$

Step 3: Equate the Exponents

Since the bases are the same (both are 10), we can equate the exponents. This is a crucial step in solving exponential equations. If $a^m = a^n$, then m = n. Applying this to our equation, we get:

k - 2 = -3

Step 4: Solve for k

Now, we have a simple linear equation in terms of 'k'. To isolate 'k', we can add 2 to both sides of the equation:

k - 2 + 2 = -3 + 2

k = -1

Therefore, the value of 'k' that satisfies the equation is -1.

Conclusion

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In this article, we embarked on a journey to find the value of 'k' in the product of powers equation $10^{-3} imes 10 imes 10^k = 10^{-3}$. We began by establishing a firm understanding of exponents and powers, exploring their fundamental concepts and the rules that govern their interactions. We then applied these rules, particularly the product of powers rule, to simplify the equation and isolate 'k'.

By equating the exponents and solving the resulting linear equation, we successfully determined that k = -1. This exercise highlights the importance of mastering exponent rules for simplifying expressions and solving equations in mathematics. With a solid grasp of these concepts, you can confidently tackle a wide range of problems involving exponents and powers.

Remember, practice is key to mastering any mathematical concept. Try solving similar problems to reinforce your understanding and build your problem-solving skills. With dedication and a clear understanding of the rules, you can confidently navigate the world of exponents and powers.

Answer

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The value of k is -1 which corresponds to option B.

B. -1