Question

Evaluate the exponential expression. Write fractions in simplest form $$11 \cdot 11^{-1}$$

Short answer

The value of the exponential expression \(11 \cdot 11^{-1}\) is 1.

Step-by-step solution

Concept of Negative Exponent

The key to understand this is knowing that any base (except zero) raised to the power of -1 equals the reciprocal of the base. So \(11^{-1} = 1/11\).

Calculations Following the Order of Operations

Now that we have rewritten the expression, the next step is multiplication, according to the PEMDAS/BODMAS rule, which governs the order of operations. Hence, \(11 * (1/11)\) equals 1, because any number multiplied by its reciprocal equals 1.

Key concepts

Negative Exponent

Understanding negative exponents is essential in simplifying exponential expressions. A negative exponent indicates that the base number is on the wrong side of the fraction and must be inverted. For example, when we see something like \( a^{-n} \), this is equivalent to \( 1/a^n \). It's a way of expressing division without writing a fraction immediately. It's important to remember that any nonzero number to the power of a negative exponent can be transformed into a positive exponent by taking the reciprocal of the base. Thus, \( 11^{-1} \) simply becomes \( 1/11 \). This property of negative exponents helps in simplifying expressions and solving equations involving exponential terms.

Simplifying Fractions

Simplifying fractions means to reduce them to their simplest form where the numerator and the denominator have no common factors other than 1. This process often involves dividing both the top and bottom numbers by their greatest common divisor (GCD). For example, if we have a fraction like \( 36/60 \), the GCD of 36 and 60 is 12, so dividing both by 12 leaves us with \( 3/5 \), which is the fraction in its simplest form. In the context of our exercise, since 11 is a prime number, the fraction \( 1/11 \) is already in simplest form because there are no common factors to divide out. Remember, simplifying fractions makes the numbers more manageable and the math easier to understand and solve.

Order of Operations

The order of operations is a fundamental concept in arithmetic and algebra that dictates the sequence in which the operations should be performed to correctly evaluate an expression. The acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right), helps students remember this sequence. When faced with an expression, you should first evaluate any expressions in parentheses, followed by exponents, then perform multiplication and division as they appear from left to right, and finally, tackle addition and subtraction in the same manner. This rule helps to avoid errors and ensures that everyone solves math expressions in a consistent way. In our exercise, multiplication is the last operation performed following the evaluation of the exponent.

Reciprocal of a Number

The reciprocal of a number is simply one divided by that number. It is the multiplicative inverse of the number, meaning when a number is multiplied by its reciprocal, the result is always 1. For any nonzero number \( n \), the reciprocal is denoted as \( 1/n \) or \( n^{-1} \). For instance, the reciprocal of 5 is \( 1/5 \), and the reciprocal of \( -3 \) is \( -1/3 \). This concept is particularly useful in division; to divide by a number, you can multiply by its reciprocal. It is also applied in solving equations and simplifying expressions, as demonstrated in our exercise, where \( 11 \) multiplied by its reciprocal \( 11^{-1} \), or \( 1/11 \), gives 1. It's important to understand this concept as it is widely used not just in algebra but in various branches of mathematics and applied sciences.