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Number and Algebra - Exponents

Grade 11IB_AI

Review the key concepts, formulae, and examples before starting your quiz.

πŸ”‘Concepts

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The base is the number being multiplied, and the exponent (or index) is the number of times it is multiplied by itself. For ana^n, aa is the base and nn is the exponent.

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The Laws of Indices apply when the bases are the same. For example, when multiplying, we add the exponents: amΓ—an=am+na^m \times a^n = a^{m+n}.

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A negative exponent indicates a reciprocal: aβˆ’n=1ana^{-n} = \frac{1}{a^n}.

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Fractional exponents represent roots. For instance, a12a^{\frac{1}{2}} is the square root of aa, and a1na^{\frac{1}{n}} is the nn-th root of aa.

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Scientific notation (Standard Form) is a way of writing very large or very small numbers in the format aΓ—10ka \times 10^k, where 1≀a<101 \le a < 10 and kk is an integer.

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In the IB AI course, you can often solve exponential equations of the form ax=ba^x = b using your Graphic Display Calculator (GDC) by finding the intersection of two graphs or using the solver function.

πŸ“Formulae

amΓ—an=am+na^m \times a^n = a^{m+n}

aman=amβˆ’n\frac{a^m}{a^n} = a^{m-n}

(am)n=amn(a^m)^n = a^{mn}

(ab)n=anbn(ab)^n = a^n b^n

a0=1, for a≠0a^0 = 1, \text{ for } a \neq 0

aβˆ’n=1ana^{-n} = \frac{1}{a^n}

a1n=ana^{\frac{1}{n}} = \sqrt[n]{a}

amn=amn=(an)ma^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m

πŸ’‘Examples

Problem 1:

Simplify the expression (3x2y3)29xβˆ’2y\frac{(3x^2y^3)^2}{9x^{-2}y} and express your answer with positive exponents.

Solution:

  1. Apply the power of a product rule to the numerator: (3x2y3)2=32(x2)2(y3)2=9x4y6(3x^2y^3)^2 = 3^2 (x^2)^2 (y^3)^2 = 9x^4y^6.
  2. Substitute back into the fraction: 9x4y69xβˆ’2y\frac{9x^4y^6}{9x^{-2}y}.
  3. Divide the coefficients: 99=1\frac{9}{9} = 1.
  4. Subtract the exponents for xx: x4βˆ’(βˆ’2)=x4+2=x6x^{4 - (-2)} = x^{4+2} = x^6.
  5. Subtract the exponents for yy: y6βˆ’1=y5y^{6-1} = y^5.
  6. Result: x6y5x^6y^5.

Explanation:

We used the rules (ab)n=anbn(ab)^n = a^n b^n, (am)n=amn(a^m)^n = a^{mn}, and aman=amβˆ’n\frac{a^m}{a^n} = a^{m-n} to simplify step-by-step.

Problem 2:

Evaluate 27βˆ’2327^{-\frac{2}{3}} without using a calculator.

Solution:

  1. Deal with the negative exponent first: 27βˆ’23=1272327^{-\frac{2}{3}} = \frac{1}{27^{\frac{2}{3}}}.
  2. Rewrite the fractional exponent as a root: 2723=(273)227^{\frac{2}{3}} = (\sqrt[3]{27})^2.
  3. Calculate the cube root: 273=3\sqrt[3]{27} = 3 (since 3Γ—3Γ—3=273 \times 3 \times 3 = 27).
  4. Square the result: 32=93^2 = 9.
  5. Put it back in the denominator: 19\frac{1}{9}.

Explanation:

Negative exponents indicate reciprocals, and fractional exponents mn\frac{m}{n} indicate the nn-th root raised to the power of mm.

Problem 3:

A population of bacteria grows according to the formula P=500Γ—(1.2)tP = 500 \times (1.2)^t, where tt is the time in hours. Calculate the population after 33 hours and write the answer in scientific notation.

Solution:

  1. Substitute t=3t = 3: P=500Γ—(1.2)3P = 500 \times (1.2)^3.
  2. Calculate (1.2)3=1.728(1.2)^3 = 1.728.
  3. Multiply: P=500Γ—1.728=864P = 500 \times 1.728 = 864.
  4. Convert to scientific notation: 864=8.64Γ—102864 = 8.64 \times 10^2.

Explanation:

We substitute the given value into the exponential growth formula and then convert the decimal number into the form aΓ—10ka \times 10^k.

Exponents Grade 11 Notes & Examples | IB AI Maths