Problem 17 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Rewrite the number without using exponents. $$ \frac{2^{-3} \cdot 2^{-4}}{2^{-5} \cdot 2^{-2}} $$

Step-by-Step Solution

Answer

The simplified expression without using exponents is \(1\).

Step 1: Apply the exponent rule to the numerator

We have two exponents with the same base (2) in the numerator. To multiply them, add the powers: \(2^{-3} \cdot 2^{-4} = 2^{-3 + (-4)}\) Now, we get: \(2^{-3 + (-4)} = 2^{-7}\) The numerator simplifies to \(2^{-7}\).

Step 2: Apply the exponent rule to the denominator

We have two exponents with the same base (2) in the denominator. To multiply them, add the powers: \(2^{-5} \cdot 2^{-2} = 2^{-5 + (-2)}\) Now, we get: \(2^{-5 + (-2)} = 2^{-7}\) The denominator simplifies to \(2^{-7}\).

Step 3: Apply the exponent rule to the entire expression

Now we have the following expression: \(\frac{2^{-7}}{2^{-7}}\) To divide the exponents, subtract their powers: \(\frac{2^{-7}}{2^{-7}} = 2^{-7 - (-7)}\) Simplify: \(2^{-7 - (-7)} = 2^{0}\)

Step 4: Rewrite the expression without exponents

Any non-zero number raised to the power of 0 is 1: \(2^{0} = 1\) The expression without exponents is 1.

Key Concepts

Exponent Rules

Understanding the rules for handling exponents is essential for simplifying complex mathematical expressions. These rules allow us to manipulate powers in algebraic terms effectively. When multiplying two terms with the same base, as seen in our exercise, we simply add their exponents. In mathematical form, \(a^m \cdot a^n = a^{m+n}\). Conversely, when dividing terms with the same base, the exponents are subtracted: \(\frac{a^m}{a^n} = a^{m-n}\). Another important rule to remember is that any number raised to the zero power is always 1, which brilliantly simplifies \(2^0 = 1\) in our final step.

Applying these exponent rules leads to efficient simplification, but be cautious! Always double-check to ensure that the base numbers are identical before applying the rules.

Negative Exponents

Negative exponents can sometimes cause confusion, but they follow a straightforward principle. A negative exponent indicates that the base is on the opposite side of a fraction line. In other words, \(a^{-n} = \frac{1}{a^n}\) and vice versa. It may help to think of negative exponents as instructions to 'take the reciprocal of the base raised to the positive exponent'.

For example, in our original problem, \(2^{-3}\) simply means \(\frac{1}{2^3}\). This understanding allows us to turn negative exponents into positive ones, often making it easier to simplify expressions. It's a subtle transformation that can dramatically simplify the algebraic expression at hand.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables (like \(x\), \(y\)), and arithmetic operations. Simplifying an algebraic expression means reducing it to the simplest form while retaining its original value. This process typically involves combining like terms, applying exponent rules, and sometimes factoring.

As you tackle algebraic expressions with exponents, remember to apply the rules consistently and watch for opportunities to transform the expression into something more manageable. For instance, turning a negative exponent into a positive one, as we have discussed, can change the entire layout of an algebraic expression, paving the way for further simplification.

Mathematical Simplification

Mathematical simplification is all about making expressions as concise as possible without altering their value. This includes reducing fractions, eliminating complex exponents, and simplifying radical expressions. Simplification is not just about aesthetics; it is crucial for solving equations, evaluating expressions, and understanding the underlying relationships between numbers and variables.

In our example, we simplified a potentially confusing fraction of exponents by using the rules for exponents, turning the entire expression into a more familiar number: 1. A keen sense of simplification can often make the difference between an intractable problem and a manageable one, which is why mastering these concepts is so important.