Question

Divide the following powers of 10 $$10^{-2} \div 10^{-4}$$

Short answer

The result of dividing \(10^{-2} \/ 10^{-4}\) is \(100\).

Step-by-step solution

Recall the Quotient of Powers Property

The Quotient of Powers Property states that when dividing two powers with the same base, you subtract the exponents: \( a^m \/ a^n = a^{m-n} \), where \( a \) is the base and \( m \) and \( n \) are the exponents. In this case, the base is 10.

Apply Quotient of Powers Property

Following the property, subtract the exponent in the denominator from the exponent in the numerator: \( 10^{-2} \/ 10^{-4} = 10^{-2 - (-4)} \).

Simplify the Exponent

Perform the subtraction in the exponent: \( -2 - (-4) = -2 + 4 = 2 \). So, the expression simplifies to \( 10^2 \).

Calculate the Result

Since \( 10^2 = 10 \times 10 = 100 \), the final result is \( 100 \).

Key concepts

Negative Exponents

Understanding negative exponents is critical when dealing with various mathematical concepts, especially when simplifying expressions. A negative exponent tells us that we should take the reciprocal of the base number and then apply the exponent as if it were positive. For instance, when you see an expression like
\(10^{-2}\),
this implies that you should take the reciprocal of \(10\) which is \(1/10\), and then raise it to the power of \(2\). Therefore, \(10^{-2} = 1/10^2 = 1/100\).
This is not immediately obvious, but once you understand the pattern, it becomes a simple process to follow. Working with negative exponents is essentially a way of expressing division within the framework of exponents.

Exponent Subtraction

Exponent subtraction is a fundamental part of simplifying expressions that involve powers of the same base, as seen in the Quotient of Powers Property. When you divide these like bases, you subtract the exponent of the denominator from the exponent of the numerator. If both exponents are negative, remember to keep track of the signs during subtraction.
In the case of \(10^{-2} \/ 10^{-4}\), you are essentially performing the operation \(-2 - (-4)\), which results in adding the opposite of \(-4\), leading to \(-2 + 4 = 2\). It is essential to respect the order of operations and properly handle negative signs to avoid errors.
When simplifying these types of expressions, keeping the base the same and directly subtracting exponents allows for a clear and straightforward solution to the problem.

Powers of Ten

The powers of ten play a unique role in our number system and are heavily utilized in scientific notation for expressing both very large and very small numbers. A power of ten simply means using ten as the base and raising it to an exponent. For positive exponents, each power of ten represents the number ten being multiplied by itself a certain number of times. For example, \(10^3 = 10 \times 10 \times 10 = 1000\).
For negative exponents with ten as the base, the principle of the denominator described in the negative exponents section applies, but with an easy pattern to follow: the number of zeros in the denominator will be equivalent to the absolute value of the exponent. Thus, a power like \(10^{-5}\) results in a fraction represented by 1 followed by a decimal point and five zeros: \(0.00001\). Powers of ten are indispensable for efficiently writing and calculating with very large or small numbers.

Simplifying Expressions

Simplifying expressions involves several techniques, including applying the properties of exponents and performing basic operations like addition and subtraction. The goal is to write the expression in the simplest form possible without changing its value. This process often includes breaking down complex expressions into more recognizable parts, combining like terms, and reducing fractions.
In the context of the problem \(10^{-2} \/ 10^{-4}\), simplifying the expression includes using the Quotient of Powers Property to subtract exponents and perform necessary computations to arrive at a final, simpler expression: \(10^2\), which can then be easily calculated as \(100\). Simplification not only makes expressions easier to read but also easier to use in further calculations and applications.