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Calculations and transformations

This page is part of the section “USE basic level. Calculations and transformations”. Here are solutions to typical problems on operations with powers, roots, logarithms and trigonometric expressions. All problems are written independently, with neat integer answers.

Problems with solutions

Problem 1. Properties of powers

Calculate:

25·2326

Apply the properties of powers: when multiplying, the exponents are added; when dividing, they are subtracted:

25+326=2826=28−6=22=4

Answer: 4.

Problem 2. Properties of roots

Calculate:

50·2

The product of roots equals the root of the product:

50·2=50·2=100=10

Answer: 10.

Problem 3. Logarithms

Calculate:

log216+log39

By the definition of the logarithm:

log216=4,log39=2
4+2=6

Answer: 6.

Problem 4. Table values of trigonometric functions

Calculate:

4·sinπ6·cosπ3

Use the table values:

sinπ6=12,cosπ3=12
4·12·12=1

Answer: 1.

Frequently asked questions

How to solve problems on powers from the USE basic level?

Use the properties of powers: a^m · a^n = a^(m+n), a^m / a^n = a^(m−n), (a^m)^n = a^(mn), a^(−n) = 1/a^n. Reducing to a common base greatly simplifies the calculations.

How to calculate the product of roots?

By the property √a · √b = √(a·b). For example, √50 · √2 = √100 = 10. If the radicand is a perfect square, extract it.

How to calculate a logarithm without a calculator?

Ask: to what power must the base be raised to get the argument? For example, log₂16 = 4, because 2⁴ = 16. log₃9 = 2, because 3² = 9.

Which table values of trigonometric functions should be known?

For the USE basic level, the values for 0°, 30°, 45°, 60°, 90° are sufficient: sin 30° = 1/2, cos 60° = 1/2, sin 45° = cos 45° = √2/2, tg 45° = 1. These values should be memorized.

See also