Review Exercise 1 — Class 9 Mathematics Solutions

Solution

1. Identify the radicand.

   The number under the radical is $7$, which is a prime number and not a perfect square.

2. Apply the radical property of real numbers.

   The square root of any positive integer that is not a perfect square has a non-terminating and non-recurring decimal representation:

   $$\sqrt{7} \approx 2.6457513...$$

   Since it cannot be expressed in the form $\frac{p}{q}$ where $p, q \in \mathbb{Z}$ and $q \neq 0$, $\sqrt{7}$ is an irrational number.

Solution

1. Identify the mathematical constants.

   • $\pi$ represents the ratio of a circle's circumference to its diameter ($\pi \approx 3.14159...$).
   • $e$ (Euler's constant) is the base of natural logarithms ($e \approx 2.71828...$).

2. Classify their decimal expansions.

   Both $\pi$ and $e$ are transcendental numbers with non-terminating, non-repeating decimal expansions. Therefore, both are irrational numbers.

Solution

1. Recall the definition of square roots.

   If $n$ is a perfect square (e.g. $4, 9, 16$), then $\sqrt{n}$ is an integer and hence rational.

2. Analyze non-perfect squares.

   If $n$ is not a perfect square, its decimal form is non-terminating and non-recurring, which means $\sqrt{n}$ cannot be written as $\frac{p}{q}$. Thus, $\sqrt{n}$ is an irrational number.

Solution

1. Analyze the sum of surds.

   Both $\sqrt{3}$ and $\sqrt{5}$ are distinct irrational numbers.

2. Verify by contradiction.

   Assume $x = \sqrt{3} + \sqrt{5}$ were rational. Squaring both sides:

   $$\begin{aligned} x^2 &= (\sqrt{3} + \sqrt{5})^2 \\ &= 3 + 2\sqrt{15} + 5 \\ &= 8 + 2\sqrt{15} \\ \sqrt{15} &= \frac{x^2 - 8}{2} \end{aligned}$$

   If $x$ were rational, then $\sqrt{15}$ would also be rational, which is a contradiction since $15$ is not a perfect square. Hence, $\sqrt{3} + \sqrt{5}$ is irrational.

Solution

1. Review equality properties.

   • Reflexive Property: $\forall x \in \mathbb{R}, \; x = x$.
   • Symmetric Property: $x = y \implies y = x$.
   • Transitive Property: $x = y \text{ and } y = z \implies x = z$.
   • Trichotomy Property: Exactly one of $x < y$, $x = y$, or $x > y$ holds.

2. Conclude the property.

   The equation $x = x$ describes the reflexive property of equality.

Solution

1. Review order properties of real numbers.

   When an inequality relationship transfers from $a$ to $b$, and from $b$ to $c$, resulting in $a > c$, it is the transitive property of inequality.

Solution

1. Express all bases as powers of $2$.

   $$\begin{aligned} 8 &= 2^3 \\ 64 &= 2^6 \end{aligned}$$

2. Substitute and simplify exponents.

   $$\begin{aligned} 2^x \times (2^3)^x &= 2^6 \\ 2^x \times 2^{3x} &= 2^6 \\ 2^{x + 3x} &= 2^6 \\ 2^{4x} &= 2^6 \end{aligned}$$

3. Equate exponents and solve for $x$.

   $$\begin{aligned} 4x &= 6 \\ x &= \frac{6}{4} \\ x &= \frac{3}{2} \end{aligned}$$

Solution

1. Review equality properties.

   The property stating that if $a = b$, then $b = a$ is the symmetric property of equality.

Solution

1. Factorize radicands into perfect square components.

   $$\begin{aligned} \sqrt{75} &= \sqrt{25 \times 3} = 5\sqrt{3} \\ \sqrt{27} &= \sqrt{9 \times 3} = 3\sqrt{3} \end{aligned}$$

2. Add the like surds.

   $$\begin{aligned} \sqrt{75} + \sqrt{27} &= 5\sqrt{3} + 3\sqrt{3} \\ &= (5 + 3)\sqrt{3} \\ &= 8\sqrt{3} \end{aligned}$$

Solution

1. Apply difference of squares identity $(a + b)(a - b) = a^2 - b^2$.

   $$\begin{aligned} (3 + \sqrt{5})(3 - \sqrt{5}) &= (3)^2 - (\sqrt{5})^2 \\ &= 9 - 5 \\ &= 4 \end{aligned}$$

2. Classify the result.

   The integer $4 = \frac{4}{1}$ is an even composite number, which is a rational number ($4 \in \mathbb{Q}$).

Solution

1. Evaluate the Left Hand Side ($\text{L.H.S.}$).

   $$\begin{aligned} \text{L.H.S.} &= a(b + c) \\ &= \frac{3}{2} \left( \frac{5}{3} + \frac{7}{5} \right) \\ &= \frac{3}{2} \left( \frac{5 \times 5 + 7 \times 3}{15} \right) \\ &= \frac{3}{2} \left( \frac{25 + 21}{15} \right) \\ &= \frac{3}{2} \left( \frac{46}{15} \right) \\ &= \frac{3 \times 46}{2 \times 15} \\ &= \frac{138}{30} \\ &= \frac{23}{5} \end{aligned}$$

2. Evaluate the Right Hand Side ($\text{R.H.S.}$).

   $$\begin{aligned} \text{R.H.S.} &= ab + ac \\ &= \left( \frac{3}{2} \times \frac{5}{3} \right) + \left( \frac{3}{2} \times \frac{7}{5} \right) \\ &= \frac{15}{6} + \frac{21}{10} \\ &= \frac{5}{2} + \frac{21}{10} \\ &= \frac{5 \times 5 + 21 \times 1}{10} \\ &= \frac{25 + 21}{10} \\ &= \frac{46}{10} \\ &= \frac{23}{5} \end{aligned}$$

3. Conclusion.

   $$\text{L.H.S.} = \text{R.H.S.} = \frac{23}{5}$$

Solution

1. Evaluate the Left Hand Side ($\text{L.H.S.}$).

   $$\begin{aligned} \text{L.H.S.} &= (a + b)c \\ &= \left( \frac{3}{2} + \frac{5}{3} \right) \times \frac{7}{5} \\ &= \left( \frac{3 \times 3 + 5 \times 2}{6} \right) \times \frac{7}{5} \\ &= \left( \frac{9 + 10}{6} \right) \times \frac{7}{5} \\ &= \left( \frac{19}{6} \right) \times \frac{7}{5} \\ &= \frac{19 \times 7}{6 \times 5} \\ &= \frac{133}{30} \end{aligned}$$

2. Evaluate the Right Hand Side ($\text{R.H.S.}$).

   $$\begin{aligned} \text{R.H.S.} &= ac + bc \\ &= \left( \frac{3}{2} \times \frac{7}{5} \right) + \left( \frac{5}{3} \times \frac{7}{5} \right) \\ &= \frac{21}{10} + \frac{35}{15} \\ &= \frac{21}{10} + \frac{7}{3} \\ &= \frac{21 \times 3 + 7 \times 10}{30} \\ &= \frac{63 + 70}{30} \\ &= \frac{133}{30} \end{aligned}$$

3. Conclusion.

   $$\text{L.H.S.} = \text{R.H.S.} = \frac{133}{30}$$

Solution

1. Evaluate the Left Hand Side ($\text{L.H.S.}$).

   $$\begin{aligned} \text{L.H.S.} &= a + (b + c) \\ &= \frac{4}{3} + \left( \frac{5}{2} + \frac{7}{4} \right) \\ &= \frac{4}{3} + \left( \frac{5 \times 2 + 7 \times 1}{4} \right) \\ &= \frac{4}{3} + \frac{17}{4} \\ &= \frac{4 \times 4 + 17 \times 3}{12} \\ &= \frac{16 + 51}{12} \\ &= \frac{67}{12} \end{aligned}$$

2. Evaluate the Right Hand Side ($\text{R.H.S.}$).

   $$\begin{aligned} \text{R.H.S.} &= (a + b) + c \\ &= \left( \frac{4}{3} + \frac{5}{2} \right) + \frac{7}{4} \\ &= \left( \frac{4 \times 2 + 5 \times 3}{6} \right) + \frac{7}{4} \\ &= \frac{23}{6} + \frac{7}{4} \\ &= \frac{23 \times 2 + 7 \times 3}{12} \\ &= \frac{46 + 21}{12} \\ &= \frac{67}{12} \end{aligned}$$

3. Conclusion.

   $$\text{L.H.S.} = \text{R.H.S.} = \frac{67}{12}$$

Solution

1. Evaluate the Left Hand Side ($\text{L.H.S.}$).

   $$\begin{aligned} \text{L.H.S.} &= a \times (b \times c) \\ &= \frac{4}{3} \times \left( \frac{5}{2} \times \frac{7}{4} \right) \\ &= \frac{4}{3} \times \frac{35}{8} \\ &= \frac{4 \times 35}{3 \times 8} \\ &= \frac{140}{24} \\ &= \frac{35}{6} \end{aligned}$$

2. Evaluate the Right Hand Side ($\text{R.H.S.}$).

   $$\begin{aligned} \text{R.H.S.} &= (a \times b) \times c \\ &= \left( \frac{4}{3} \times \frac{5}{2} \right) \times \frac{7}{4} \\ &= \frac{20}{6} \times \frac{7}{4} \\ &= \frac{10}{3} \times \frac{7}{4} \\ &= \frac{70}{12} \\ &= \frac{35}{6} \end{aligned}$$

3. Conclusion.

   $$\text{L.H.S.} = \text{R.H.S.} = \frac{35}{6}$$

Solution

1. Convert radical form to exponential form.

   Using $\sqrt[n]{a} = a^{\frac{1}{n}}$:

   $$\begin{aligned} \sqrt[5]{\frac{x^{15} y^{35}}{z^{20}}} &= \left(\frac{x^{15} y^{35}}{z^{20}}\right)^{\frac{1}{5}} \end{aligned}$$

2. Distribute exponent $\frac{1}{5}$ to all numerator and denominator factors.

   $$\begin{aligned} &= \frac{(x^{15})^{\frac{1}{5}} (y^{35})^{\frac{1}{5}}}{(z^{20})^{\frac{1}{5}}} \end{aligned}$$

3. Multiply exponents using $(a^m)^n = a^{mn}$.

   $$\begin{aligned} &= \frac{x^{15 \times \frac{1}{5}} \cdot y^{35 \times \frac{1}{5}}}{z^{20 \times \frac{1}{5}}} \\ &= \frac{x^3 y^7}{z^4} \end{aligned}$$

Solution

1. Express $27$ as $3^3$.

   $$\begin{aligned} \sqrt[3]{(27)^{2x}} &= \sqrt[3]{(3^3)^{2x}} \\ &= \sqrt[3]{3^{6x}} \end{aligned}$$

2. Convert cube root to exponential form.

   $$\begin{aligned} &= (3^{6x})^{\frac{1}{3}} \\ &= 3^{6x \times \frac{1}{3}} \\ &= 3^{2x} \quad (\text{or } 9^x) \end{aligned}$$

Solution

1. Expand powers with sum exponents using $a^{m+n} = a^m a^n$.

   $$\begin{aligned} \frac{6(3)^{n+2}}{3^{n+1} - 3^n} &= \frac{6 \cdot (3^n \cdot 3^2)}{(3^n \cdot 3^1) - 3^n} \end{aligned}$$

2. Factor out $3^n$ from numerator and denominator.

   $$\begin{aligned} &= \frac{6 \cdot 9 \cdot 3^n}{3^n (3 - 1)} \end{aligned}$$

3. Cancel common factor $3^n$ and evaluate.

   $$\begin{aligned} &= \frac{54 \cdot 3^n}{2 \cdot 3^n} \\ &= \frac{54}{2} \\ &= 27 \end{aligned}$$