Test Your Brain: Solve These 12 Maths Questions Without a Pen

Method 1: Step-by-Step BODMAS (Simplifying the Brackets First)

Step A: Resolve Parentheses/Brackets first. Look inside the round brackets: \(33 + 27\). Let's calculate this simple sum first:

\[33 + 27 = 60\]

Step B: Rewrite the expression. Substituting 60 back into the expression leaves us with:

\[60 \cdot 4 + 6\]

Step C: Multiplication has priority. We have a multiplication (\(\cdot 4\)) and an addition (\(+ 6\)). Since multiplication is higher in precedence than addition, we must do this part first:

\[60 \cdot 4 = 240\]

Step D: Final Addition. Now, perform the last addition to get the answer:

\[240 + 6 = 246\]

Method 2: The Distributive Law (The Smart Classroom Trick)

The Distributive Property allows us to expand the multiplication across terms inside a bracket: \(a(b + c) = ab + ac\). Let us verify our answer using this mathematical principle:

Step A: Distribute the factor of 4. Multiply 4 by both numbers inside the parentheses:

\[(33 \cdot 4) + (27 \cdot 4) + 6\]

Step B: Calculate individual products. Perform the multiplications:

\[132 + 108 + 6\]

Step C: Sum the terms from left to right. First, add 132 and 108:

\[240 + 6 = 246\]

Method 1: Isolation using Inverse Operations

Step A: Subtract 5 from both sides. We want to eliminate the constant \(+5\) on the left. So we subtract 5 from both sides:

\[7x^2 = 180 - 5\]\[7x^2 = 175\]

Step B: Divide both sides by 7. Now we isolate \(x^2\) by dividing both sides by the multiplier 7:

\[x^2 = \frac{175}{7}\]\[x^2 = 25\]

Step C: Take the square root of both sides. Remember, any positive real number has two square roots (a positive and a negative one):

\[x = \pm\sqrt{25} \implies x = 5 \quad \text{or} \quad x = -5\]

Method 2: Factoring as a Difference of Squares

Step A: Move all terms to the left-hand side. Subtract 180 from both sides:

\[7x^2 + 5 - 180 = 0 \implies 7x^2 - 175 = 0\]

Step B: Extract the common numerical factor. Divide the entire equation by 7:

\[7(x^2 - 25) = 0 \implies x^2 - 25 = 0\]

Step C: Apply the difference of squares identity. Recall that \(a^2 - b^2 = (a - b)(a + b)\). Here, \(25\) can be written as \(5^2\):

\[(x - 5)(x + 5) = 0\]

For this expression to equal zero, either the first bracket must be zero, or the second must be zero. This gives us our two roots: \(x = 5\) or \(x = -5\).

Method 1: Direct Algebraic Multiplication

Step A: Multiply both sides by 5. Since \(a\) is divided by 5 on the left, we perform the inverse operation to isolate it:

\[a = \frac{36}{100} \cdot 5\]

Step B: Calculate the product in the numerator. Multiply 36 by 5:

\[a = \frac{180}{100}\]

Step C: Simplify the fraction to a decimal. Divide by 100:

\[a = 1.8\]

Method 2: Simplifying the Right-Hand Fraction First

Step A: Simplify the fraction \(\frac{36}{100}\). Divide the numerator and denominator by their greatest common divisor, which is 4:

\[\frac{36 \div 4}{100 \div 4} = \frac{9}{25}\]

Our equation now becomes simpler:

\[\frac{a}{5} = \frac{9}{25}\]

Step B: Scale the denominators. Notice that \(5 \times 5 = 25\). To make the denominators equal, we scale the left fraction by multiplying both top and bottom by 5:

\[\frac{5a}{25} = \frac{9}{25}\]

Step C: Equate the numerators. Since the bases are identical, the tops must be equal:

\[5a = 9 \implies a = \frac{9}{5} = 1.8\]

Method 1: Direct Combination of Like Terms

Step A: Combine terms on the left. Subtract the terms directly as they share the exact same variable part (\(b^2\)):

\[(7 - 3)b^2 = 16 \implies 4b^2 = 16\]

Step B: Divide both sides by 4. Isolate the quadratic variable:

\[b^2 = \frac{16}{4} \implies b^2 = 4\]

Step C: Take the square root. Solve for \(b\):

\[b = \pm\sqrt{4} \implies b = 2 \quad \text{or} \quad b = -2\]

Method 2: Factoring and Division

Step A: Take the common factor \(b^2\) outside brackets.

\[b^2(7 - 3) = 16\]\[b^2(4) = 16 \implies 4b^2 = 16\]

Step B: Isolate the variable. Shift the multiplier 4 to the other side as a division term:

\[b^2 = 4\]\[b = 2 \quad \text{or} \quad b = -2\]

Method 1: Step-by-Step Value Replacement

Step A: Evaluate the first root \(\sqrt{64}\). We look for a positive integer that multiplies by itself to give 64:

\[8 \cdot 8 = 64 \implies \sqrt{64} = 8\]

Step B: Solve the zero exponent term \(7^0\). In index laws, any non-zero real number raised to the power of 0 is mathematically defined as 1:

\[7^0 = 1\]

Step C: Evaluate the denominator \(\sqrt{9}\). Find the square root of 9:

\[3 \cdot 3 = 9 \implies \sqrt{9} = 3\]

Step D: Reconstruct the fraction. Substitute these simple numbers back into the expression:

\[\frac{8 + 1}{3} \implies \frac{9}{3} = 3\]

Method 2: Splitting the Fraction

We can also split the single fraction into two smaller fractions before simplifying:

\[\frac{\sqrt{64}}{\sqrt{9}} + \frac{7^0}{\sqrt{9}}\]

Substitute our known square roots and index values:

\[\frac{8}{3} + \frac{1}{3}\]

Since the denominators are identical, simply add the numerators:

\[\frac{8 + 1}{3} = \frac{9}{3} = 3\]

Method 1: Direct Value Substitution

Step A: Find individual root values. We solve the terms in the numerator and denominator:

\[\sqrt{81} = 9 \quad (\text{since } 9 \cdot 9 = 81)\]\[\sqrt{16} = 4 \quad (\text{since } 4 \cdot 4 = 16)\]

Step B: Resolve the exponent term. Any base with a power of 0 is simplified to 1:

\[5^0 = 1\]

Step C: Substitute and calculate. Place these back into our expression:

\[\frac{9 - 1}{4} \implies \frac{8}{4} = 2\]

Method 2: Splitting and Simplifying

Let us split the terms to double-check our arithmetic calculation:

\[\frac{\sqrt{81}}{\sqrt{16}} - \frac{5^0}{\sqrt{16}}\]\[\frac{9}{4} - \frac{1}{4} = \frac{9 - 1}{4} = \frac{8}{4} = 2\]

Method 1: Inside-Out Calculation

Step A: Solve the numerator. We calculate each square root independently:

\[\sqrt{25} = 5\]\[\sqrt{9} = 3\]\[\text{Numerator} = 5 - 3 = 2\]

Step B: Solve the denominator. Here, the subtraction sits inside the radical, meaning we must perform the subtraction first, before applying the square root:

\[25 - 9 = 16\]Now we calculate the root of the result:\[\text{Denominator} = \sqrt{16} = 4\]

Step C: Divide the terms. Divide the simplified numerator by the denominator:

\[\frac{\text{Numerator}}{\text{Denominator}} = \frac{2}{4} = 0.5 \quad \left(\text{or } \frac{1}{2}\right)\]

Method 2: Proving the Algebraic Inequality

Let us mathematically demonstrate why we cannot distribute subtraction inside square roots. In algebra:

\[\sqrt{A} - \sqrt{B} \neq \sqrt{A - B}\]

If we substitute our values (\(A = 25, B = 9\)):

\[\sqrt{25} - \sqrt{9} = 5 - 3 = 2\]\[\sqrt{25 - 9} = \sqrt{16} = 4\]

Since \(2 \neq 4\), this proves that roots cannot be combined during subtraction or addition. It is a mathematical impossibility!

Method 1: Aggregation of Variables

Step A: Combine like terms. We have two identical \(m^3\) terms on the left side:

\[2m^3 = 16\]

Step B: Isolate the cube. Divide both sides of our equation by 2:

\[m^3 = \frac{16}{2} \implies m^3 = 8\]

Step C: Solve for \(m\) by taking the cube root. We find a number which, when cubed, yields 8:

\[2 \cdot 2 \cdot 2 = 8 \implies m = \sqrt[3]{8} = 2\]

Method 2: Factoring out the Cube

Let us pull the common factor \(m^3\) outside of parenthetical terms:

\[m^3(1 + 1) = 16 \implies 2 \cdot m^3 = 16\]

Divide by 2 to isolate the cube:

\[m^3 = 8 \implies m = 2\]

Method 1: Inner Simplification first (The Direct Way)

Step A: Calculate the square roots inside brackets.

\[\sqrt{64} = 8\]\[\sqrt{16} = 4\]

Step B: Simplify the resulting fraction. Substitute these integers:

\[\frac{8}{4} = 2\]

Step C: Apply the outer exponent. Square the simplified result:

\[(2)^2 = 4\]

Method 2: Distributive Exponent Law

Recall the exponent rule for quotients: \(\left(\frac{x}{y}\right)^n = \frac{x^n}{y^n}\). Let us apply this:

Step A: Distribute the square to the numerator and denominator.

\[\frac{(\sqrt{64})^2}{(\sqrt{16})^2}\]

Step B: Squares and square roots cancel each other. Since they are inverse operations, we get:

\[\frac{64}{16}\]

Step C: Perform the division. Let's simplify the fraction:

\[\frac{64}{16} = 4\]

Method 1: Basic Evaluation

Step A: Resolve \(\sqrt{9}\). We know that \(\sqrt{9} = 3\).

Step B: Substitute into the expression. Place 3 into each root's position:

\[\left( \frac{3 + 3}{3} \right)^2\]

Step C: Simplify inside parenthetical boundaries. Perform the addition and division:

\[\left( \frac{6}{3} \right)^2 \implies (2)^2 = 4\]

Method 2: Variable Substitution (The Genius Trick)

We can solve this without evaluating the root at all! Let us define a temporary variable \(y = \sqrt{9}\):

Step A: Rewrite the fraction with variable \(y\).

\[\left( \frac{y + y}{y} \right)^2\]

Step B: Simplify the numerator. Add \(y + y\):

\[\left( \frac{2y}{y} \right)^2\]

Step C: Cancel out the common variable \(y\). Since \(y\) exists on both the top and the bottom, they divide out completely:

\[(2)^2 = 4\]

This means no matter what positive number is placed inside the roots, as long as they are all identical, the final output will always be 4!

Method 1: Cross Multiplication

Step A: Multiply across diagonal lines. We draw an imaginary cross and multiply terms:

\[x \cdot x = 9 \cdot 4\]\[x^2 = 36\]

Step B: Take the square root of both sides. Remember, quadratic variables yield two distinct solutions:

\[x = \pm\sqrt{36} \implies x = 6 \quad \text{or} \quad x = -6\]

Method 2: Rational Term Scaling

Let us scale both fractions to have a common denominator of \(4x\):

Step A: Multiply top and bottom of the first fraction by 4.

\[\frac{36}{4x}\]

Step B: Multiply top and bottom of the second fraction by \(x\).

\[\frac{x^2}{4x}\]

Step C: Set the numerators equal. Since they now share a common denominator of \(4x\) (where \(x \neq 0\)):

\[36 = x^2 \implies x = 6 \quad \text{or} \quad x = -6\]

Method 1: Logical Analysis of Exponent Laws

Is there any exponent that makes all non-zero bases yield the exact same value? Yes, the exponent of zero!

Recall that \(A^0 = 1\) and \(B^0 = 1\) for all positive real bases.

Step A: Set the shared exponent to zero. For the two sides to balance and equal 1, the exponent must be zero:

\[x + 8 = 0\]

Step B: Solve for \(x\). Subtract 8 from both sides of the equation:

\[x = -8\]

Method 2: Rigorous Algebraic Division

Step A: Divide both sides by the term on the right. Divide by \(6^{x+8}\):

\[\frac{5^{x+8}}{6^{x+8}} = 1\]

Step B: Combine the bases using exponent laws. Since they share a common power, we can write:

\[\left(\frac{5}{6}\right)^{x+8} = 1\]

Step C: Match the bases. We can write 1 as our base raised to the power of 0:

\[\left(\frac{5}{6}\right)^{x+8} = \left(\frac{5}{6}\right)^0\]

Since the bases are identical, their exponents must be equal:

\[x + 8 = 0 \implies x = -8\]

1. Look for Brackets: Always resolve parenthetical terms before doing outside multiplications or additions.

2. Follow BODMAS: Write down the letters B-O-D-M-A-S on your rough sheets to avoid simple ordering mistakes.

3. Exponent Zero: Keep in mind that any real non-zero base raised to a power of 0 is simplified to 1.

4. Negative Roots: Double-check if your quadratic equation solution has a negative real root as well!