In mathematics, indices provide a convenient way to denote repeated multiplication. Commonly referred to as powers or exponents, they become pivotal when working in algebra, calculus, and various other domains of math.
Basic Rules of Indice
1. Multiplication with the same base:
When multiplying two terms with the same base, the base remains unchanged, and their powers (indices) are added together.
Rule:
\(a^m \times a^n = a^{m+n}\)
Example 1:
\(3^2 \times 3^3 = 3^{2+3} = 3^5 = 243\)
Example 2:
\(x^4 \times x^5 = x^{4+5} = x^9\)
2. Division with the same base:
When dividing two terms with the same base, the base remains unchanged, and the exponent of the denominator is subtracted from that of the numerator.
Rule:
\(\frac{a^m}{a^n} = a^{m-n}\)
Example 1:
\(\frac{5^4}{5^2} = 5^{4-2} = 5^2 = 25\)
Example 2:
\(\frac{y^7}{y^3} = y^{7-3} = y^4\)
3. Power of a power:
When an exponential expression (a number with an exponent) is raised to another power, you multiply the exponents together.
Rule:
If \((a^m)^n\), then it can be written as \(a^{m \times n} \)
Explanation:
Raising an exponent to another power means you’re multiplying the base by itself multiple times.
Example 1:
\((2^2)^3 = 2^{2 \times 3} = 2^6 \)
This results in multiplying 2 by itself six times, equaling 64.
Example 2:
\((x^3)^4 = x^{3 \times 4} = x^{12} \)
This represents \(x\) multiplied by itself twelve times.
4. Negative powers:
Any term with a negative exponent can be represented as the reciprocal of the term with the corresponding positive exponent.
Formula: \(a^{-n} = \frac{1}{a^n}\)
Example 1: \(5^{-2} = \frac{1}{5^2} = \frac{1}{25}\)
Example 2: \(m^{-3} = \frac{1}{m^3}\)
Activities:
1. Evaluate and Simplify: \(4^3 \times 4^4\)
Hint: When multiplying two numbers with the same base, add their exponents.
\(a^m \times a^n = a^{m+n}\)
E.g. \(2^3 \times 2^2 = 2^{3+2} = 2^5\)
E.g. \(x^4 \times x^3 = x^{4+3} = x^7\)
2. Evaluate and Simplify: \(\frac{6^5}{6^2}\)
Hint: For division with the same base, subtract the exponent of the denominator from the exponent of the numerator.
\(\frac{a^m}{a^n} = a^{m-n}\)
E.g. \(\frac{5^4}{5^2} = 5^{4-2} = 5^2\)
E.g. \(\frac{y^6}{y^4} = y^{6-4} = y^2\)
3. Evaluate and Simplify:
\((3^2)^3\)
Hint: When a power is raised to another power, multiply the exponents.
\((a^m)^n = a^{m \times n}\)
E.g. \((x^2)^3 = 2^{2 \times 3} = 2^6\)
E.g. \((a^3)^4 = a^{3 \times 4} = a^{12}\)
4. Evaluate and Simplify:
\(y^{-4} \times y^6\)
Hint: When multiplying two numbers with the same base, add their exponents.
\(a^m \times a^n = a^{m+n}\)
E.g. \(3^{-2} \times 3^5 = 3^{-2+5} = 3^3\)
E.g. \(m^{-1} \times m^3 = m^{-1+3} = m^2\)
5. Evaluate and Simplify:
\((n^3)^2\)
Hint: When a power is raised to another power, multiply the exponents.
\((a^m)^n = a^{m \times n}\)
E.g. \((4^2)^2 = 4^{2 \times 2} = 4^4\)
E.g. \((z^5)^2 = z^{5 \times 2} = z^(10)\)
Tricky Exam-Style Questions:
1. If \(2^{x+1} = 32\), determine x.
Rule Applied: In this case, we need to express 32 as a power of 2, then compare the exponents.
More on this: \(2^{y+2} = 16\), determine y.
Another one: \(a^{n+3} = a^8\), determine n.
2. Simplify: \(\frac{a^3 \times a^4}{a^5}\).
Rule Applied: First, combine the powers using multiplication rule and then use the division rule.
More on this: Simplify \(\frac{3^2 \times 3^3}{3^4}\)
Another one: Simplify \(\frac{b^4 \times b^5}{b^8}\)
3. Given \(b^{m-2} \times b^{3m+1} = b^{17}\), find the value of m.
Rule Applied: Combine the exponents using the multiplication rule and equate to given exponent.
More on this: Given that \(2^{x-3} \times 2^{4x+2} = 2^{18}\), find x.
Another one: Given that \(c^{n-1} \times c^{2n+2} = c^{14}\), find n.
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